Native here needs to be planted in December - dormant.  I wouldn't use rye as a cover in that scenario at all.  I would've done beans then planned on planting after Thanksgiving.  Then figured on getting iffy stands cuz i've not had the best luck with native but it usually comes along after a few years.  Let the weeds/grass mix grow and just mow it.  Never seen native look good first few years but kinda gets it's act together 3-5 years in. 

Oats are less allelopathic than rye.