Well while in college almost 40 years ago i had a class in botany...bear with me...
in the class we were asked to section into microscopic thin sections about 15 microns thick, some tree stems (about 1 inch thick) well the knife would not cut thru because of the chemical content of the wood, so we soaked the pieces in pure ethylene glycol for about a week and you could just about tie them into a knot after that....then we ran the section thru the staining process (soaking them in various grade of alcohol 10%, 20%, 40 %, 60%, 80%, 99%, then the aniline dye. Well ethlene glycol is just a chemical that the alcohol likes to take back out of the wood.
Maybe i should tell you ethylene gylcol is very lethal and i do not know if it is legal to buy...it was back then, because there were not so many nut cases running around threatening everybody...and nobody cared about future tax payers.
I know auto antifreeze is mainly ethylene glycol but is dyed and has other crap in it so it is not useable for this.
Maybe while the wood veneer would be flexible you could sand bag or vacuum it the like to the surface with appropriate glue.
Then lay absorbent towels soaked with various grades of alcohol work up to 99% to get the glycol out until you have a paintable surface....
This would be something Tom Anderson could tell use since he does something to bend his thick maple tops on his 'drop top' models...but i doubt he is going to reveal anything to us!
I dunno if this might work, I am not endorsing it, it is a bad chemical and will waste your dog or cat or child in a day if they drink it, let alone if you absorb it some into you skin ( i think it destroys your kidneys) back then the EPA did not exist...so let this be your caveate emptor!