Measuring current on the RF side of the choke and expecting a "good" signal is not reliable. Even taking voltage readings there (coax) can load the front end and cause unpredictable results, especially on digital signals. If you can get to the traces, the best place to meter is on the regulator's (output) side of the RFC. If possible to access, cut the trace from the regulator output and get in series there. Use either of those points (ref gnd) for a voltage TP. You can always solderbridge the cut back together when you've fixed the problem.
I'd be curious to know what the control input voltage is doing during these tests. It does sound like the (TO220 case?) regulator is going into current limit mode for whatever reason. Are you certain it's the correct replacement device? Limiting can be causes by it overheating as well, or it being defective and acting as if it is overheating. Maybe you got a dud. Except for mil spec parts, new does not always mean good.
If the regulator's control input varies proportional to the "works for a bit then quits" cycles then the problem is before the regulator, possibly the driver transistor (likely another "chip" SMT 3 legged black square) is malfunctioning under load.
Best of luck getting it figured out.