Probably, but it'll be a close thing. You'll lose (at least) a bit over 1.2 volts to the regulator and a red diode will have a forward voltage of a bit less than 2 volts. Assuming a fully charged 18650 @ 4.2V you'd have a volt to spare. As the battery discharges though, the regulator will stop.....regulating.
If a '317 regulator is all that will fit in your budget you'd be better off using 2 cells. Otherwise invest a few bux in a proper driver from DTR.