Answering my own questions:

T2 to T1 connection with no onward BP and no way to get one.

Short answer - not a problem, including lounge access.
My situation:
Star Alliance arrival flight to ICN, T2. landed around 7am
Separate ticket, AA to DFW departing T1 around 6pm

The process.

Head to transfers to T1. (long walk)
Scan Passport to get access to the train to the Concourse (and later T1)
Get on train to Concourse.
Get off at Concourse. Follow the signs to T1.
Scan Passport to get access to the train to T1.
Follow the signs to connections.
Just before security, scan passport.
Go to front of security and hand passport to scan.
Clear security and you are now airside T1.

I have access to the OW lounge. So headed to the lounge and they looked at my itinerary and asked to see my FF card. (Digital)
Granted access.

Only thing I needed to do is go to the gate to get a bp before boarding.
(AA does not have a transfer desk.)

Hope this helps others.