I'm assuming this can be rapidly eliminated - but just in case:
If a UA has been in a system and broadcast a name for a certain length of time (long enough to be received by whoever), it switches to its second (ship drawing) mode. But if it is taken to a system/station where it hasn't been before it reverts to the name mode...
It's probably worth going by the assumption that these are extremely unsophisticated machines and aren't going to respond to attempts at contact. But equally, it's plausible that they may have a further behaviour mode, which kicks in after the first two....
Sorry but no.
In the test I made last night, the same UA was swapped from two different ships, same station, without changing the instance, and it changed its MORSE only according to the ship that deployed it. It did one, or the other. For a few swaps. I even jumped to another system, than back to Thompson Dock, and did it all again.