Several year CoCoRaHS user here. I use the mobile app. Muuuuuuuch easier than using their website.
FWIW - they ask for readings 'at' 7am localtime for their 7am-7am rain day, although they'll take any readings they can get. You can even enter multi-day readings (which would be particularly hard to get into weewx), and alter the time-of-day you took the reading to line up with when you actually did so. They also support you adding readings after the fact anytime you want, as well as editing old entries to clean up errors etc.
Regardless, I'd suggest the o.p. look into just writing a standalone secondary sqlite3 db first before worrying anything weewx initially. It wouldn't be too hard to programmatically generate the sqlite3 insert statement once you knew the date+time and value to use there. Schema would be super-simple, just do the minimum weewx would expect.
Perhaps something like the following....
schema = [
('dateTime', 'INTEGER NOT NULL PRIMARY KEY'),
('usUnits', 'INTEGER NOT NULL'),
('interval', 'INTEGER NOT NULL'),
('cocorahs','INTEGER'),
]
There are lots of sqlite3 examples out there but
(this one) is pretty simple for starters.
Once you have a db with data in it in a weewx-expected format, do a simple skin reading from that secondary db and see what the graphs look like for day/week/month/year etc. to see if it's worth connecting the two.
Heck, use bogus data. You can always delete your cocorahs.sdb and start over clean. That's part of why a standalone db for this might help.