I am assuming you're talking about the Tesla Type II charging (cable), not the Tesla wall charger (Type III).. I would think the UK cable is not functionally different than the USA cable, it is not uncommon for companies to use a single cable for global (just with different connector adapters for different countrys' receptacles), so to save costs in design. Also global cable being able to handle 50Hz or 60Hz, the 220v, 230v, and 240v are very similar and I would think the cable is designed to input 208v-250v like they imply in this doc:
...Where the doc says 240v recommended, but then they show the NEMA 14-50 plug/receptacle as voltage 208v-250v.. Now I cannot say if the cables for say USA and UK are the same or different, but I would almost rather pick the one I knew would for sure handle the Hz frequency for my country (in case it matters or if they are different).
Again, I would guess that they might use a single cable for both countries, but I couldn't confirm without seeing specs sheets for both country's cables. It might be that the car circuit is just the one looking for the 208v-240v 50 or 60 Hz signal.
I would think you could run that inverter you got at auction, but you might have to turn down the input amps charging setting on the Tesla car low enough to not trip the inverter.
Quick example, when I wired my brother's Tesla to a dryer type plug in his garage (we used a 30a circuit for this), we had to turn his car (on the dashboard setting) down to 29a, so it wouldn't trip the 30a breaker.
The Tesla doesn't need a split phase, I remember when I wired his Tesla plug in his old house garage, we only had 3 wires (two hot and a ground) and the Tesla charged fine. At his new house garage we had a 4-wire supply so we went ahead and wired the 4th neutral wire into the receptacle outlet, but it really doesn't matter as far as the Tesla cable input is concerned (doesn't need neutral when charging on the higher voltage).
Make sure you're using suitable gauge cable and protect it with a breaker.
Does your inverter say if it is 50 or 60 Hz? You might look into whether the car specs say if you can take both 50 or 60 Hz or if it matters. Also check your inverter to see if the output voltage is user selectable, you might be able to change it to 240v output, but if not, 230v should be fine. I would just want to make sure the car doesn't care about the Hz if the inverter cannot be changed, to me that would be more important to know about than voltage being exactly right on.