Thanks for those suggestions, ledtester. I understand that a power supply may not duplicate a battery's response, but for testing the constant current circuit, which is the big issue in doubt, I'll just be firing two 8us bursts 18us apart (38KHz, 30% duty cycle). I think a honking big capacitor should be enough to prevent any sag. Then if I get that working, I can do a reality test with a battery.
I decided to go ahead and order the cheap bench supply, mainly because I think it's time I had one. But in the meantime, I think I can use the LM317 supply with the 16V feed. It should tolerate the short test bursts ok. And as you say, I can always use a pass transistor with the LM317 if I can find one in my junque box.