For any reasonable ematch, you need 300-800 milliamps for a second or so. I've flown the same 9V battery for ten or more flights and not gone below 9V on the cell. Lipos can produce lots more current than this application needs; so much that for some altimeters you run the risk of blowing up the switching MOSFETs.
A 1S lipo very quickly dips below 4V (they spend most of their time at about 3.8V) so you are hoping that the regulator in the SL has enough margin to operate over temperature and load. Which it probably does, but...
My opinion is based on my experience with using them specifically with the S100 and SLCF, as well as using a 9volt with a failure. The failure was due to the alkaline's, relative, inability to recovery quickly if you happen to fire back to back. If you have a SL100 or SLCF with some flights, check the logs and post the result showing the impact the charge has on the voltage. In most cases it will drop immediately after the charge, whereas with a Li-Po remains the same.
I also fly redundant e-matches, so 2 e-matches per channel and I have yet to get a flight back with less than 4.1 volts remaining on my SL100 or SLCF. 3.7-3.8 is actually about a 50% charge for a 1s Li-Po so if that is what you are getting then I would say that the battery was not fully charged to start with. My testing and flights have been extreme when it comes to Li-Pos. I have literally connected a fully charge, 4.2 volts, 1S Li-Po to an SL100 with a featherweight mag switch, turned it on to run the diagnostics, turned it off and left if for a couple days to see the draw or net effect of the mag switch, then proceeded to run 2 back to back both main and drogue tests with dual e-matches and the voltage never went below 4 volts.
I agree that a Li-Po can damage the MOSFETs that is why it is important to not over specify the size, hence my recommendation for a 1s over a 2s. This is not an opinion but factual, below is a cut and paste, (modified for this thread) from another forum member who responded to my thread on this subject back in May 2014.
"Current fires a pyro and current destroys FETs. Since ohm law always applies, the more voltage you shove through the pyro circuit, the more current it will draw. Using one cell instead of 2 means 2 times less voltage and therefore 2 times less current draw. If your e-matches work with 1 cell battery than you should go that route. If it doesn't then try a 2S battery which is also what a lot of the "9 volt transistor battery style" LiPos are."