That's the same thing
@txcomp used in his design so he could use drill batteries or whatever. The FX Chronograph really needs a regulated voltage source since it is so finicky. Even the Alkaline aren't ideal, but at least they decay through the working range longer than the Li-Ion as you have found. You could also look for a "boost" regulator that will just step up the voltage to stay at 4.1V or whatever until the regulator drops out. You will probably get more run-time from a "buck" converter as you were referring to.
As
@nervoustrig mentioned, that circuit will protect your Chronograph. You could also use a "Positive Temperature Coefficient" (or self-resetting) fuse.
Something to keep in mind too is that some switching regulators have a minimum current required to regulate, and some if this is not supplied, have the ability to run away. I'm saying this because you may not be able to just put a DVM on the output of the regulator and get the right voltage. It may be reading what your battery is putting out, and still be working correctly. You just need to draw a few mA for it to regulate properly.
Good luck, and thanks for sharing your findings.