We do part-time as well, but it seems to be becoming more and more full time as time goes on.
We started with DS at 2.5 months. I'm lucky because he cued for poops right from the outset (that's why I started EC, in fact bc it was so obvious!) so we started catching most of those right away. So I haven't been in the same position as you for poops, so not sure if this is helpful or not...DS usually poops after eating. This is pretty normal as eating gets the bowels going. Maybe try this: If DD is "due" for a BM (it's been a day or whatever the "average" interval is) take her potty shortly after she eats and cue for a BM and give her a good 3 or 4 minutes to relax and go, cuing every 10 or 15 seconds. Not sure if this will work or not, maybe she'll get the idea. DS went 3 times a day consistently, but his schedule did change (went down to once a day, now back to twice a day). If she doesn't signal for poops and doesn't have a consistent schedule it'll make it hard to catch them consistently. Do you use a probiotic? I know DS really got on a "schedule" once we started giving him a probiotic daily.
As for diaper free time, we did this early on but don't really do much anymore. We did diaper free time when I was trying to learn his cues. I've decided DS doesn't cue for pees (after much careful study!! Maybe I'm just missing the obvious!) So then we went the prefold with no cover route as well. This worked well bc I could tell as soon as DS peed or even while he was peeing. If I caught him while peeing I'd cue. If it was right after, I'd say "Duncan went pee-pee" and change his diaper right away. So I think using prefolds with no cover is fine.
I agree with Perdita, just go with the flow, offer as much as you can, and DD will catch on and you'll start having fewer misses and more catches.