I'm currently having a go at the same thing and asked advice a few weeks ago. Milliput is good. I couldn't get the putty to stay in place with the cling film, but ended up using ut on top to stop my hand sticking to it. I've had trouble from the new putty coming off the grip due to original oil finish acting as non-stick coating, then using small blobs of blue tak for now to stick it back on. I wish I'd spent more time with blue tak to understand what I wanted to achieve and have wasted quite a bit of putty.
Ersonally I don't rate normal wood filler as it's so soft you just squeeze it out when gripping the gun.
Finally the setting time disappears very quickly, do be clear what you are aiming to achieve first
Best of luck
Andy
Steyr Evo 10, Anschutz 8001, BSA Scorpion SE, BSA light Pattern, HW77k, HW30s, AA TX200, HW45 Blackstar, HW40, Walther CP88, Webley Premier E, TBC!