Actually looks like a pretty easy job.
I'm not an electrician. I would not take my advice. I'd just listen to what
@eprosenx says - ignore me.
Personally, if it's a short term thing, I'd try this:
1) Unless there are open slots I'm missing in your pictures (really can't see the bottom of the bus bars), you need to do something like this: Replace 4 of those wide breakers with 2 tandem breakers (convert 4 breaker slots to two slots with 4 breakers - this does mean moving some circuits to the opposite 120V phase so just be aware if anything is using powerline communication...can cause problems) to make room for a 50A 2-pole breaker (double the width of a single breaker). Sounds like you're already familiar with this since you said there was plenty of room. I guess you have to lose your AFCI breakers (looks like that is what they are) if you go with tandem (my understanding is this is no great loss...though totally not to code)?
This step might be totally unnecessary - you said there was plenty of room so maybe you can just add the 50A breaker somewhere to the bus bars which is hidden in your last couple pictures.
2) Cut a small exploratory hole in the drywall below the panel near one of the studs. I'd aim to make the hole relatively high; you don't want the outlet too close to the floor - ideally the UMC will be able to hang down
without touching the ground - you have the UMC so you can test fit the height.
3) Make sure it is free and clear run up to the breaker box. If obstructed, evaluate whether to continue (drill hole through offending cross-brace 2x4 or whatever - technically if you go through a 2x4 you have to have a nail guard plate installed on the 2x4 behind the drywall though I think - so hopefully no drilling needed). I'm concerned because I see that join in the drywall going horizontally; not sure what that join in the drywall is for...
4) If all is well, make the appropriate larger sized hole in the drywall and add a decent depth metal 2-gang junction box (have to make sure it can fit
the 14-50 depth) somehow without making a mess of it, and attach it firmly to the stud. Some drywall repair might be required depending on how much you can fit through the hole. There's a lot of details here I am glossing over; there are a dizzying array of options as far as junction boxes go - I have no idea what might be best. Wear robust gloves around these boxes, to avoid slicing your hands open. They're so sharp, don't ask how I know.
5) Remove an appropriate size (3/4"?) knockout (those perforated circles - they will knockout quite easily) from the bottom of the panel. (You could also do this first instead of making holes in the drywall...but then you have a hole in the bottom of the panel if you can't proceed.) Get the appropriate clamps and such (you'll need two of appropriate diameter - 3/4"? - they're threaded to allow clamping to the knockout hole and the junction box, and then have screws and a clamp to clamp the wire) to secure the Romex cable out of the panel and into the new junction box, and run a short run of 6/3 Romex (maybe you can get some brand new scrap from somewhere/someone on eBay, or buy it by the foot somewhere) from your new breaker to the box. Make sure to ground the junction box with a grounding screw and a little piece of the grounding copper wire from the Romex - really probably unnecessary, since it will be grounded by the outlet, but is required by code.
6) Get a Hubbell or Bryant 14-50 outlet (they're better & safer than the Leviton ones), connect to the Romex using the appropriate torque, and install along with a matching plate. Be sure to properly strip the wires and don't clamp insulation by mistake. Be sure that all wires are connected appropriately and securely both in the panel and in the outlet box.
Be sure to make the ground pin on the top. That way the UMC will just hang.
Then use your UMC with a 14-50 adapter (if you didn't get one from Tesla you have to buy it from them). You might want to rig up something so the cable is off the floor so you don't run it over by mistake.
I don't recommend the above. I'm not an electrician. I could be wrong, I might have forgotten something. It might not be to code. You could start a fire if you do it wrong. You have to know what you're doing and you have to make everything properly tight, to avoid fires. But it's what I would try, if I wasn't installing a Wall Connector.
I think the total costs should be max $100 with the breakers (depends how many you need), outlet, bits and pieces, and a short piece of 6/3 Romex from somewhere for cheap.
Or you could get an electrician to do the same...which has its own risks. Or just take advice from
@eprosenx - he probably has better ideas with a better handle on the terminology and the best way to do this.
(I have a temporary Tesla charging station with SOOW cable that I just run out the front of the panel with the screws not all the way down, but that is for ultra short term use and is certainly not safe or to code - it has an outdoor 14-50 box on the end with the receptacle)
