Will my solanum faxum (album) come back from frost damage?

welcome to GoY.
try scraping the bark with your thumb nail. if it is green below the bark it is still alive and hopefully it will come from the base. If all brown below the bark check all stems. brown usually means dead but dont be too hasty to dig it out.

Mine took a hammering too but is showing green under the bark so there is hope

So did mine - I cut it back by two-thirds and there are little new shoots appearing now. Have a look - it may well be OK.

Good luck - they are lovely climbers!