"This is more of a family question.
I've noticed that Max has both good and bad times with the sleeping. Things had been improving, a lot, particularly with the daytime napping, and then they tanked. Then they got better, then worse, etc., and I've been trying to monitor what's going on in the house that may explain this. I finally figured it out - things get bad when my husband is around more during the day - weekends or university holidays (thus, the sleep disaster that was late December).
I'm Max's primary caretaker and while I'm not rigid with a schedule, I do things a certain way and Max and I seemed to be getting into a groove. Paul does things differently, and I want to respect that as much as possible, since there's clearly more than one way to skin a cat. However, some things that Paul does result in poor or no daytime naps and I pay for that at night. Plus we end up with a grumbly baby, and that's no fun.
I don't want to usurp Paul, and I don't want to imply that my way is "right" or "more right." And I really rely on the time off from baby-care that Paul gives me during those times for my own sanity. Can Max adjust? Do I try to force Paul into doing things my way?
P.S. Paul and I typically communicate really well, but I'm sensitive on this. I got a lot more confidence in my ability to handle Max before Paul did...he's still a little wavery, and I don't want to shake that at all."
There should be a name for this common syndrome. How about Other Parent Disruption Factor? (Not that you moms and dads who go off to work all day are other. We know you're equal parents who are completely capable of caring for your children just as well as those of us who are the primary caregivers do. But "other parent" is just easier to say and type than "non-primary caregiver" is and less derogatory than "secondary caregiver" is.)
I don't know a single family that doesn't have or hasn't had some form of this problem. I think yours is more intense because Max is still so young and his sleep isn't really that solid anyway. He may also just be a sensitive kid, which is wonderful except on this particular topic. But OPDF seems to be one of those common things that no one talks about before it happens.
The thing that made it vastly better for us is something I don't really recommend: My husband got laid off from his job. So he was home all the time (he and I both did freelance work and just traded the boy back and forth) and became part of the normal routine. For the 15 months he didn't have a full-time job he was also a primary caregiver, and that helped make the OPDF almost non-existent.
Except, of course, that my husband has different energy than I do. So the routines were the same, and El Chico doesn't sleep any differently after a day with his dad, but there are things he does with his dad that he'd never try with me and vice versa. Sometimes my husband and I will compare notes and we're just stunned at how differently El Chico (and now El Pequeño) act when they're alone with him vs. alone with me.
People (or at least me) talk all the time about how babies learn very quickly what the rules are with different people. The classic example is that grandparents are notorious for letting kids have all sorts of foods that they never get to have at home, and the kids learn not even to expect those foods at home. But I think kids also sense different expectations and energies from different people and respond to those.
The upshot is that I'm not sure there's anything that you could do about Max's reaction to Paul, even assuming you wanted to. The problem doesn't really seem to be that Max is different with Paul, but that it's ending up being your problem because Max has crappy nights after being with Paul all day. The good news is that the more time Max and Paul spend alone together, the more confident Paul will get and the less Max will have his sleep affected by Fun Day With Daddy. The bad news is that this won't happen in the next few weeks, probably.
I wonder if you and Paul could try to troubleshoot the routine to see if there's anything Paul could tweak while he's alone with Max. It's not about making Paul do things your way--it's about deciding together that it's important for a 9-month-old to nap, so the nap must be respected. How Paul gets him down for the nap has nothing to do with you, and you won't even care about it, as long as a decent nap happens. If there are other things that contribute to bad sleep (like Daddy-Max Dance Party or something similar that gets Max all riled up), those could switch to morning or very early afternoon so they aren't still having an effect by bedtime.
If the nap is a given (and, yes, a 9-month-old does need a nap every day, and most of them need two naps at that age), then it's really not your business how it happens, and Paul and Max will work that out. So I'd focus less on the ethics and emotion of forcing your way on Paul, and just make it a stated value that whoever has Max on any given day respects the naps. Then when you come home and find them doing some strange trust exercise game involving Max launching himself off the couch onto Paul's chest, you won't even have to care because you'll know at least he had a nap earlier.