Hi s7horton
It's hard to beat a true and tried way of doing it by using contact cement, as you know the key is to get it to stick quick without any air pockets and using a roller to get it flat and without the need of clamps to hold it down.
Titebond II is great stuff BUT it's not contact cement and you may need to clamp the stock in place so it can setup,it takes air to get it hard so to speak and you just blocked it with the veneer not to say anything about the air pockets that you will need to roll out.
I would take it out side or to the garage and put on the veneer then take it back down stairs after a hour or two.
Or put a fan in the window and pull out the odors in the basement.
It would be a shame if the veneer started to come up after a bit, that's always a drag to redo it or redo the corners the 1st time something hits it and it pops up or snaps off.
Just a small NOTE***
I'm sure you know that 3M puts out great spray contact cement .it's quick and easy to use unlike the mess with paint brush or the wipe and spread on way.
I use the 3M #77 all the time and it holds just about anything.
All you need is just some sticks,3M,and a roller and your done.
Putting a HOT iron on wood will get the wood filbers to come up and you may need to resand the base or to say redo the hole thing.
You will always have moister in wood,it's just the way wood is
not to say anything about the danger of burn marks or staning the top..
But this is just my 2 cents
Good Luck
Bj