All good suggestions, and some interesting tips too.
I'm not a huge fan of track bouncing, tho' I've done plenty of it. It has it's utility in production, especially in the analog world with limited track counts.
With these (or any) two machines, there are many variables and varied approaches you could take to everything. Probably a different method u could exploit for e'day of the week.
Bear in mind at it's most basic level, there is no such thing as a lossless bounce, (sorry digital guys!)
However, losses can be minimized and fidelity can be preserved.
Any "Internal" bounce operation will be a "destructive" bounce when carried out, by design of the technique. Bounce tracks are fixed in the mix (on the target tracks), and the source tracks for the bounce are taped over, destructively rewriting tracks as the method progresses thru production. An "External" bounce operation preserves the source tracks from the primary bounce on it's own media. There's an advantage there if you consider revisiting the primary bounce in the longevity of a project, if it comes to that your options are open.
I've done both internal "destructive" bounces and external bounces, & I generally prefer external bounces FWIW. However, I might be more inclined to bounce a lesser track in back of a stronger or more prominent part, played live-to-track (with off-tape backing). That's maybe more like "bouncing-lite". I don't want to get too many tangents about track bouncing.
With these two different machines, there are several viable ways you could set up for tracking, bouncing, etc., and there's certainly no "best" way or even a concensus on any single thing.
The recognizable differences in fidelity between the TSR8 and 388 should be a non issue, as long as you're bouncing to the higher fidelity device, i.e., from the 388 to the TSR8.
I'm wholeheartedly in favor of adding parts live-to-track during the bounce. Been there & done that. I give it thumbs up as bouncing techniques go.
As far as sychronization being offered up as a solution, IMO is way above a better choice than bouncing,... with reservations that it's a very fiddly and complicated layer of technology to lay on top of the entire scheme of your studio setup. You'd be at minimum DIY'ing the cabling, which is not always a slam dunk or simple proposition when dealing with obsolete hardware. Not impossible, but a major PITA. Setup of the sync'd ATR system is fiddly at best that may need a few tweaks, but when set up properly and worked out it presents a great alternative to bouncing. Way better.
[Specifically for the TSR8 sync to the 388 a Tascam ATS-500 and IF-500 would be needed, as the TSR8 sync is serial and the 388's is parallel I/O. I know little or nothing about what other sync devices might work, if any.]
2 words on DIY cabling & sync PITA: sweet beats.
Tho' Beck's advice is all sound, (heh: pun), at a certain point between our varied tips you're splitting hairs. Either way you're creating 14 "virtual" tracks thru some level of bouncing. I say take an entier 8 tracks mixed off the 388 & dub them down to 2 tracks in stereo onto the TSR8, then knock yourself out on the remaining 6 tracks, (total 14). Beck says record 6 & bounce to 2 tracks in stereo, record 4 & bounce to another 2 (stereo bounce), (by then is 10 parts crammed into 4 tracks in a 1st Gen bounce), then knock yourself out on the remaining tracks, (presumably recording on them as primary tracks) for *total 14. However, if you take it to the hilt, there's (3+2+1) more (parts) "virtual" tracks you can record thru more bouncing. (Actually, the last track is not bounced, but it's the logical culmination of the technique).
All track bouncing (or any other technique) can be taken too far at times. I happen to think the "maximum" internal bounce, aka "reduction bounce" on a single tape deck may sound better on paper than in practice. I'm most in favor of a non-destructive external bouncing technique over an internal destructive bouncing technique. Sorry to repeat,... but the paradox is I've probably done more internal/destructive bounces than external. (Probably why I dislike it more).
With that being said, I think track bouncing has some utility in the studio. There are different ways to approach it and levels of intensity of the technique. Sorry to have hit a tangent over track bouncing! All the varied aspects of track bouncing was not part of the question. Heh.
Using 2 quality 8-tracks as the 388 & TSR8 could produce some great results in combination thru bouncing.
Aye!