You have a Rule of Six main panel. That's a problem.
A long time ago, 100A breakers were prohibitively expensive. So they came up with the "Rule of Six" - which said your so-called "main breaker" could actually be made of up to six smaller breakers. So there'd be a (typically 60A) breaker going to a subpanel for smaller loads, and then (up to five) other 240V breakers powering range, A/C, water heater, dryer etc. Your panel is a "Rule of Two" lol, but it's operating under the same old rule. Good news? No.
The problem with the old Rule of Six approach - and the reason it was outlawed - is that the breaker handles often added up to larger than the service size. So it was very common to have 40A A/C, 50A range, 30A dryer, 30A water heater, 60A small load panel, and we're at 210A already -- all on a 125A service! How did that work? How was that safe?
By requiring a NEC Article 220 Load Calculation. This uses fudge factors (like 3 VA per square foot for general loads) and discounting (like reducing nameplate draw of ranges/ovens to account for them cycling on/off) to arrive at a realistic safe current draw. This is science-based and it works fine.
The problem is People adding things. They are supposed to re-do the Load Calculation, but lol nobody does. And so over the years, "Rule of Six" panels are often dangerously overloaded. That's why they've been banned.
"But Harper... If I get a 400A service, that comes as dual 200A breakers. So they're still using the Rule of Six, clearly." Yes, that is true - having >1 breaker isn't the issue; it's breaker handles that add up to more than the service size.
So you can see where just changing a 40A main breaker to a 40/50 is just cruising for bruising.
You're getting mixed advice from the electricians because you're talking to them by phone, so they aren't understanding the Rule of Six weirdness at work here. They'd get onsite and go "Oh wait..."
So you find out your service size. Then...
The next step is to find out from the power company what service size they have given you. That matters because of allocation of transformer capacity, but it also decided the service drop wires they used from the pole. And it also decided your service entrance wires from your weatherhead through the meter into the main panel. You can't exceed that.
By "not exceed it" I mean one of two things. Either
- You do a Load Calculation on the house, with the EVSE added, and see if you are still below the service size. If so, you can do it "the old fashioned way" Rule of Six style. (note that this Load Calculation is required in any case, given the size of the addition). OR
- conform to the "breaker handles <= service size" quasi-rule. That eliminates the worst of the hazard right off the bat.
Absent any additional information, my only advice would be to use a 50/20 on a breaker inside the house - taking care to place it so you don't exceed 125A stab limits (for both breakers across from each other). Your installer did the right thing putting all large breakers on one side, to minimize the stab limit problem, so I'd replace two 20A singles on the left side with a 20/50/20.
A few other thoughts
If your state has adopted NEC 2020, as of the date on your permit, don't use a receptacle. Hard-wire the EVSE. The reason is NEC 2020 requires GFCI on 240V outlets, and that's stupid, since the EVSE already has a vastly superior GFCI built into it. (it self-resets, resolving most nuisance trips, and can warn your phone of persistent problems). Learn how to do the junction box wiring if a replacement EVSE is needed in a hurry.
If you need a socket because you're planning to use the provide "lump in the cord" travel charger as your daily EVSE, forget it. That's for opportunity charging when out and about, it should live in the trunk with your "Tesla to J1772" adapter. (those 2 plugs are electrically compatible; the adapter is just wires).
Current is a soft setting for EVSEs. So if it's advertised as a 50A unit, you can make it a 20, 30, 40 or even 60A unit by flipping some internal switches. Keep that in mind if you get in a jam.
If a 2nd EV is in your future, learn all about Share2 power-sharing before adding any more circuits.
40A is way too big for an air conditioner. Look at getting a newer one that is 20-30 SEER (they go as high as 38 SEER and can run in reverse to give you very efficient heating in spring and fall). It will require less power, which will help. SEER is BTUs per watt.
You can make similar load shaves by going to a heat pump dryer (which are good for your HVAC bill too, as they are NOT ejecting large volumes of conditioned air). Or a heat pump water heater (warning: will "air condition" your water heater site whether you want it or not!)
If the Square D panel is reasonably recent, it might be possible to get those last 4 spaces functional. You'd need to identify the panel's age and "Box Size", then shop for a Square D 24-space panel with that same Box Size and series. Buy the panel whole, note that it has basically 3 parts: the cover, "box" and a huge assembly that contains everything else; this assembly unbolts with 3-4 bolts at most. Don't disassemble it unnecessarily. Make sure all the mounting bolts and slots are the same. Remove all your neutral wires, pop all your breakers off (don't detach wires from breakers!) and swap bus assemblies. Make sure everything fits properly and shipshape, no ugly hacks. (NEC 110.12).
Note that I plan to charge only at night when family is sleeping with low load on the power supply.
But if your house is well insulated, it may be smart to run your A/C then. Anyway, Code doesn't allow "I promise" as a method; after all you may sell your home to someone else who doesn't know to do that. Electricity is for using, users shouldn't have to be careful like that.
We're a few years out from affordable Smart Panels which solve problems like this for you. (Span will do it today, if cost is no object. Span even has a partner EVSE which will dynamically adjust charge rate to stay within house energy limits).
Note that I plan to charge only at night when family is sleeping with low load on the power supply.
