You’ll have a hard time trying to get the right measurements from an old stretched, sticky belt.
All you need is the Tascam part number, which is 5800698201, to order from Tascam. But for your records, if you have a local source here are the dimensions in inches:
Circumference: 13.15
Width: 0.29
Thickness: 0.03
As for tape thickness, tapes vary in total thickness within a class like 1-mil or 1.5-mil. A 1-mil tape can be as much as 1.5-mil and some 1.5-mil tapes are over 2-mil in total thickness. Generally speaking, if the tape is 1800 ft long you have 1-mil tape.
Those in the 1-mil class that are good for the 388:
Ampex/Quantegy 407 & 457
3M/Scotch 207
Maxell XLI 35-90B (UD 35-90 in a pinch, but it has no backcoating)
EMTEC/RMGI LPR35.
There is at least one exception to the rule. EMTEC or RMGI SM468 does well on machines made for 1-mil tape and is one of my favs.
7" Ampex won't have a date code on the box. As I mentioned before I could tell you by looking at the front of the box if you post a pic. The reels are good for other tape as JP mentioned (but it's a pain) or for the empty reel itself as a take-up reel. The Ampex you have should be the nice smoke color reels.