Tesla Mobile Connector tripping GFCI on NEMA 5-15
1 - If this is a 20A circuit but the device is designed for 15A (two straight up prongs instead of one up, one sideways) then changing to a 20A receptacle will make absolutely no difference.
2 - GFCI tripping, in some circumstances, is due to an actual failure of the GFCI itself. I don't think 6 years old is all that old (I have some that are 20 years old and working just fine), but replacing it may have some effect. A GFCI in a garage is likely subject to extreme temperatures which can affect lifetime.
3 - GFCI tripping, in most circumstances, is due to a failure of the device you are plugging in or due to moisture. Make sure that the plug/cord is totally dry before plugging it in and that there are no exposed conductors, crimped cords, etc. that could be the source of a real problem.
4 - Charging a Tesla, or any EV, off a 5-15 receptacle is extremely limiting. Going up to a 5-20, provided the charger will actually make use of it (which should mean that it has a 5-20 plug, which is actually quite unlikely because of other better options) would increase charge rate by 33%. But going to a 6-20 (20A 240V) would more than double your charge rate and going up from that to a 14-50 (50A 240V) will give you a LOT more charging capacity. See the Tesla manual for details.
Upgrading the receptacle (beyond 5-20) will require new wiring. How easy or hard that will be depends on:

Your total electric service - i.e., whether you have enough capacity or need to get a utility upgrade
Space on the panel for a new double-breaker
Work involved running a new circuit from your panel to the garage

But long term, it will almost certainly be worth the effort. It can easily make the difference between being able to charge overnight after a long trip vs. getting up in the morning and finding you are only at 25%.