@FreeMan answered your question on what to do (and a bit of why) so I will provide some follow-up info about "why" and various things to consider.
I'm not an electrician, but the National Electric Code (NEC) (lawbook for electricians) can be one of your best friends (assuming you can read the fancy and verbose language).
To be code compliant, at least one light switch needs a dedicated neutral. The purpose of this, as stated by the 2017 NEC code book, is so that the extra wire may be used in the future if the property owner decides to replace regular manually switched lights with motion sensor lights (or any type of automatic switch that requires a neutral). Since the locations of your two switches on opposite sides of your kitchen probably don't cumulatively have a line of sight to every square inch of your kitchen floor, per 404.2.C you would need to have the neutral wire at both switches.
You should also be aware that cramming too many wires into too small a box is a code violation. There are formulas for calculating the minimum size of a box for the number of wires you plan on putting in it (look up "equivalent conductors" and "sizing an electrical box" to learn more), but it's honestly probably fine to eyeball. Logically, too many wires = too great a chance that one wire getting tugged accidentally tugs on another wire, loosening both wires and creating a potential electrical fault situation (whether an arc fault, shorting fault, or a stray hot wire with voltage and an uncapped metal end). Additionally, more wire = more heat = more danger (though probably not as major a concern as the electrical faults). Due to this, you may want to move your new pass-through-the-box wiring outside the box itself, making sure to support it by industrially stapling it (or using nail-in fasteners) to a stud (unmoving wooden plank) as you run the wire. Be extremely careful to not fray the wires in the securement step as that could lead to any of the later-mentioned faults, which could start fires or create hazardous fumes until the breaker trips (if the breakers function correctly).
Here is part of the 2017 NEC code book, specifically in reference to "Switches" (the relevant part) near the top of the document, and "Receptacles, Cord Connectors, and Attachment
Plugs (Caps)". Section 404.2 is highly relevant to you. 404.2.A basically says to not have a switch on the neutral ("grounded conductor"), which your diagram complies with; the reason is for unnecessary voltage - all your hot connections are still energized (have dangerous voltage) despite being switched off! Here are several diagrams that explain how you would think to do it if it's your first time, and then how to correctly implement a dedicated neutral (since this is a webarchive link, it will take probably at least 5 seconds for the images to load and maybe a few refreshes).
Listening to @FreeMan's advice would yield a dedicated neutral (assuming I read his intentions correctly). Also, their advice about having a straight run of wire that is not spliced at boxes C5 and C1 (since those top two wires are not intended to be powering nor returning power from those boxes specifically) but instead goes through those two boxes is perfect.
Whenever wiring something, always try to imagine not having the electrical plans accessible (they could be lost) and trying to make changes to the wiring in the future. Opening a box (a lighting can's box in your case) with an extra, unused hot and neutral splice may lead future-you to think that it may have been used for something else, like another light in the above floor. That could lead to adding more lights on a circuit that was never designed for that much amperage. The neutral going from the last can on the run (C5) straight to the switch that receives always-hot power might make you think "maybe the neutral is the hot and the installer just used the wrong-color wires and didn't reidentify the white to black".
Thinking about future work is also the reason behind not having your neutral go from C5 to S1, but instead having your neutral end at C5, taking the C5->C4->C3->C2->C1->S2->C1->C5->S1 route. It's generally easier to reason about where a possible bad connection is when your return (neutral) and supply (hot) are in the same place.
EDIT: After doing some research, it turns out that TRIAC types of dimmer switches are notoriously bad with LEDs because the circuitry inside LEDs tries to account for fluctuations in power supply, so the LED circuitry stores more energy than it normally would when a dimmer limits the amount of current flowing into the LED, brightening the LED if it has enough power. Separately, does your dimmer switch have a neutral (not just two travelers plus an outgoing hot plus ground)?
The neutral connection provides the essential return path for the dimmer even when the load is disconnected or is in a state where it does not draw any current. It not only ensures that the dimmer has power to drive its own internal circuitry even when the load is disconnected or off, it also provides a clean signal of the incoming AC-power source for detection of zero-crossings and synchronization with the line. This makes for stable, reliable phase-controlled dimming.
If your dimmer switch is in fact not a combo 3-way switch + dimmer switch (i.e., if it's only a dimmer switch), then you need to connect your dimmer switch directly to your house's power instead of only giving it power when the other 3-way switch is "on". Maybe that will fix your issue, but I am honestly not sure at all, so you should ask around more. The linked source includes more suggestions that should work.
- Use a voltage field tester (the "wand looking thing" officially called a Non-Contact Voltage Tester) to test if a wire is hot (has voltage, i.e. "power") or not. Be sure your wires are not too close together when using this tester because you may be accidentally testing the wrong wire.
- You can also use the probes on a multimeter instead, but you need to touch the metal probes to the metal part of the wire in this case, putting you at a higher risk for shock.
- Using the multimeter probes, you can test if there is a break in the wire by doing a Continuity Test on a dead circuit (the multimeter will supply the power, don't burn the fuses on the multimeter by supplying 120 volts); you CAN also do this test with a voltage field tester.
- Make sure all your wirenut splice connections are solid. Do a gentle pull/tug test on your wirenuts to make sure the wires inside don't come apart. Metals that barely touch each other have a VERY high resistance, leading to heat production, leading to fire; imagine trying to force 10 gallons of water per second (10Amps) down a coffee straw (a small surface area between two metal wires) - you're going to get a lot of resistance!
- Make sure you're using the correct size of wirenut for the number of wires (and the size of those wires) that you will put inside the wirenut. This info should be on the package of the wirenut.
- Pre-twist the wires with the wirenut (or pliers with a wide gripping surface) to ensure that even if the wirenut DOES fall off, the wires still remain twisted together and therefore remain electrically connected.
- LEDs are polarized, meaning they like power that flows in one direction. You're probably running your LEDs straight off AC (Alternating Current) where the LED has an internal rectifier, meaning you don't need to worry about it, instead of having an external rectifier convert AC->DC. If you haven't had to physically install a rectifier for the lights, then there's no chance you wired the poles wrong (because AC flows in both directions, not just one!).