I would go for bipolars, they are easier to configure into the flasher circuit.
The base drive should be no problem with reasonably designed flasher - it flows only when the LED is ON and it needs to be about 4% of the LED current, so not significant.
But be aware about the seemingly very simple (two transistor,...) flasher circuits found around the internet: Many rely on circuit parasitics (like battery internal resistance so won't work with modern low impedance batteries, Germanium transistors saturating rather softly so do not work that well with silicon, on a cold incandescent having way lower resistance, many to really low transistor current gain,...), many need explicite kickup, so at the end very often they tend to stay in a kind of "limbo" where the light is permanently ON or OFF and do not oscillate.
A good mental test is, what would be the hypothetical operating point of the transistors when you remove all the capacitive feedbacks. If the thing may stay or stays with some transistors fully ON or OFF.
or
), the thing won't work in a reliable way.
That equilibrium point needs to be with the circuit "somewhere in the middle" - transistors only partially conducting, neither fully ON, nor OFF. From that state the adding of the capacitors will make it unstable so oscillate.
Good example is
The output transistor just can not stay ON, because it would loose base drive...
So without the capacitor the circuit stays in a state where the voltage drop across the output transistor is about 1V or so, so perfectly in the range where it is a good amplifier and not stuck ON switch...
My only remark for operation at lower Vbat is you need to add some 100Ohm parallel to the LED (for your case, where the series resistor with LED will be around 1 Ohm or nothing, the base resistor of the power transistor about 100Ohm,...), so full battery voltage will become present across the circuit when the transistor is OFF...
