You can build Armory on 32bit Ubuntu just fine. What you cannot expect is a 32 bit version of Armory to go online. But it's fine offline. That stuff will be able to sign for BCH tx as well.
The simplest way for you to build your binary would be to run a VM of your target OS and arch on your online machine, build the binary from that, then take it to your offline signer.
Lastly, is it possible to just copy the signing wallets from the offline machine to another machine - perhaps temporarily?
Don't, you'll be compromising that wallet.
I successfully compiled in a VM but as the method of running it is "python ArmoryQT.py" (and of course ArmoryQt.py isn't a binary) I wasn't able to tell exactly how to move that to my offline machine. I looked through the python file a little, trying to find where the binary actually is but as I'm not a developer, I couldn't really reach a conclusion. How, after following the instructions here (
https://btcarmory.com/docs/building/ - Ubuntu Build Instructions) do I find the binary? I realize this is probably an incredibly ignorant question but I've Googled building dpkg and other such things to no avail.
TIA.