As far as frequency difference and motor speed, note what Ed Beal says.
Your worst case play is to get an "online" UPS from the European world. Most UPS's just pass-through AC power as they get it, while also using it to charge the battery. Only when power fails do they switch to battery+DC-AC inverter. For an online UPS, it's different. They convert all incoming power to DC, and the DC-AC inverter is used at all times - so a switch to battery is seamless and instant. As a result, it always outputs 50Hz even if the input is 60Hz. But that only matters if your motor cares. Let's see the nameplate.
Use the existing 240v outlet in my kitchen with an adapter. That would require calling an electrician to do the work.
No. That "existing 240V outlet" is for the oven/range. It has a 40A or 50A circuit breaker. That appliance is designed to fail in a way that would trip a 13A or 16A breaker. If a 40A breaker were "protecting" it, the breaker wouldn't even notice while the appliance burns.
We really need to see the nameplate on the appliance, but I'm guessing it's between 6 and 13 amps at 230V. It would take a 15A (or tops, 20A) breaker to provide adequate protection.
If you don't plan to use an electric range or oven, it is possible to change both the socket and the breaker to 15-20A, using the existing wiring in the walls. While you're at it, you might want to use something like Legrand Wiremold surface conduit to bring the socket somewhere more practical. Code requires kitchen appliances have cords no longer than 2 feet, and strongly discourages extension cords.
Keep in mind the 240V outlet will NOT be GFCI, which means if it falls into the sink, you are dead. So don't do insane things like draping an extension cord across the sink area, or having socket+cord such that it could get near the sink. However if you are one of the several US states that has adopted NEC 2020, your 240V circuit will need to be GFCI - upside no worries; downside requires $80 GFCI breaker.
Use a transformer. Based on the fact that the KM096 can draw up to 1500W, I'm thinking 3000W to be safe
No. The problem is, EU wall sockets can deliver 3000W, But USA wall sockets cannot.
The downside of our safer 120V is that less power is available, and for many EU heating appliances, that's just not enough. That's one thing that endears EU appliances to Americans, actually; they work better because they have more power to work with.
What you'll need to do is have an electrician add a dedicated 240V kitchen countertop circuit. You won't find too many electricians who will be will be willing to do this without that $80 GFCI breaker, which will require 2 full spaces in the panel. If your panel is full, that will require some rearranging which will be extra work and $$$, which may also expose other flaws which also need addressing. Upshot, this could snowball into $500, or even as much as $2000 of work if the electrician arrives and finds a hopelessly miswired FPE main panel, for instance.
You may get very lucky and find an existing "Multi-Wire Branch Circuit" configured in a way that lets the electrician swap in a dual-voltage receptacle and you're good to go. That would be an easy $100-ish job.