You'd have to figure out how the fan works and what the momentary push button does. That would determine your next course of action. Is it just a dry contact? Is it high or low voltage? AC or DC?

From what you are describing, you probably need some sort of relay. And you can decide whether you want the Raspberry Pi to operate the relay, or whether you want the Lutron switch to operate it.

Without me actually seeing what you have, it is really hard to make a good guess. Here is a conservative guess that probably will work. I would try something like https://smile.amazon.com/gp/product/B0753ZGST4 You can wire this relay so that it gets controlled by your Lutron switch, and the fan is then connected to the outputs of the relay.

Using an AC powered SSR (solid state relay) is more expensive than using a cheap DC relay that is directly operated by the Raspberry Pi. But I am suggesting this solution, as it stands a better chance of working, even if you don't fully understand how all the existing parts expect to operate. You are of course welcome to simplify things and use cheaper components, if you have decided that those will work correctly.

Since you are working with relatively low currents, you can probably even remove the heat sink from the SSR and just hide the relay inside of a junction box. If you use self-tapping screws to attach the relay to the junction box, the box itself should be a more than adequate heat sink. I have successfully used this particular SSR with a high-current water pump and even then it doesn't get hot; a bathroom fan is much lower current. The heatsink shouldn't be necessary in that case.

The problem with this approach is that the Lutron switch turns all the way on or off. But you just want instantaneous push-button operation. This is not something that Lutron does out of the box, but you can fix that with the integration API. You would write some code that observes that status of the switch and immediately turns it off again, after it has been turned on.

In other words, any time you see '~OUTPUT,<IntegrationID>,1,100.00', you write '#OUTPUT,<IntegrationID>,1,0'. As I said, Home Assistant can probably do that for you. But even a dumb little shell script should be able to handle such a simple task. It all just depends on what you are most comfortable with.

Edit: Depending on what exactly the Panasonic fan is doing, the small leakage current for the SSR could pose a problem. If that's the case, then use an old-school mechanical relay instead. A quick search on Amazon finds: https://smile.amazon.com/Power-Relay-Module-Control-Voltage/dp/B089QCG6GK This relay wouldn't be a good option to directly control a fan motor, as the contacts will probably fuse after a couple of hundred or thousand operations (switching inductive loads is tricky business), but it sounds as if the Panasonic fan is using electronic controls. And in that case, this relay should be just fine.