@Shawn Ackerman
Lots of good discussion above.
You said you are worried about setting a poor precedent. That has already been set with it being this late in the month before you are getting around to collecting rent. Even if it’s an understandable situation, the tenant doesn’t know the details of your business, just knows you didn’t get around to collecting rent until it was 20 days late.
I feel there are issues with your PM as well- couldn’t tell if this is a PM you inherited with the property or a PM that you have worked with for a long time. Non-payment of rent and partial payment is something I like to have already worked out where they don’t have to ask you each time. I prefer a standard process (i.e. they don’t pay on time, send letter, they don’t pay by XYZ, file in court).
In some jurisdictions, accepting partial payment pushes back your right to file for eviction so please check your local laws first. Again, this is something I would expect to lean on my property manager for, and I could be reading it wrong but it doesn’t sound like your PM provided you that knowledge up front (i.e. you could accept partial payment but it will push you back X weeks in eviction). Maybe this is a new jurisdiction for him/her and it takes time to get up to speed but it’s been 3 weeks that you’ve had the property.
I do have one property in a jurisdiction where accepting partial rent hamstrings me, so in that case I would be tough. Pay or quit with a late fee.
I have others where that’s not an issue, so if something like this happened there I would probably take it with a late fee and ensure (in writing) they understand the next month’s rent is due 1 Oct. My concern is, Sept is almost over and if they don't have Sept’s rent, they probably won’t have Oct's on time either. My other concern is I’ve never bought a house where the existing tenants were a dream (smile).
Look forward to hearing how things worked out!