Couple more details that might be helpful:

Autumn foliage colour is orange.

The tree started blooming in early May (I'm in the equivalent of zone 9 or 10)

I suspect it's budded on a 4-foot single trunk (standard form). The branches are very thin and they slope downwards (very elegantly) instead of growing upwards. some of the new branches start horizontal and then slope downwards as they get longer.

Check out the Catawba cultivar.

'catawba' is a good possibility. also look at 'zuni' and some of the other purples at the link below.
'zuni' should have peeling bark, with age.

Here is a link that might be useful: crape myrtle cultivars

Thank you Hoseman and Jeff. I checked both varieties not only on Jeff's link but also on other sites.

Both Catawba and Zuni seem to have an upright growth habit, but my mystery crape is weepy or cascading, so it can't be either. Mine sort of grows DOWNWARDS.