The question should have been "What YEAR should I have toured the Dolomites" ;-)
Assuming you don't have a time machine to go back 5-10 years before speed cameras popped up everywhere, speed limits got reduced and they started banning motorccles from certain roads on certain days I would say the best time is before some more power rangers fuck off the powers that be and there is even less accessible ;-)
It can be really hot mid summer (July / August) but as its the Alps it could also be really cold, mid-late June would be my preferred time before too many hordes of cyclists and walkers rock up and spoil everything.
Check Local School holiday dates, IIRC they start and end a bit later than ours.
2nd / 3rd week in September might be good, maybe a bit cooler and not too busy, slightly higher cold / snow risk, shorter days and probably a bit nippy at altitude 1st thing in morning.
Lots of climate data online, but remember climate is what you expect and weather is what you will get.
Arabba is good base.
Best for Luxury;
https://www.hotelevaldo.it/en/hotel-dolomites/1-0.html
Best for quirkynes (Part of building is old WW1 Fort)
https://www.alforte.com/en/
If going for Al Forte book a package with food, firstly its very nice, secondly it works out quite economical, thirdly there is bugger all else in walking distance, have stayed twice but in large group (30+ Bikes, we had booked the whole Hotel - and got cracking deal on that basis) so not bothered about more remote location.
Lots of other choice in Arabba and Cortina, but between me and my friends they are the only hotels in the area we have used.
Don't worry too much about lack of car park, my mate has been going there regularly for decades with large groups, so far no bikes have gone walkies overnight, pretty sure they have never had a bike nicked from anywhere, but a few never made it back on time due to getting into an argument with the famous dolomite rocks.