The pattern selection is supposed to be the same as on the UHF2150B:
In fact, I had an older copy of 13510.pdf from February 16, 2000, which I have uploaded, here:
It will take quite a bit more power to engage the relay than it will to keep the relay engaged. You can expect a higher current requirement for a small fraction of a second when engaging the relay. However, in practice, this is negligible in the grand scheme of things.
The coil of a common...
Yes, 24awg is far too small. At minimum, match the gauge of your extension wire to the gauge of the flying leads coming out of your speaker. Going even larger is okay. Get rid of the 3.5mm plug and jack.
A 537 is pretty small once you take it out of its can. Doing so can be quite tempting when a small space is all you have to work with. However, removing the can alters the thermal characteristics of the flasher and could lead to an even more lopsided flash.