I've got 2 hubs due to range limitations. They are linked with HubConnect. The server hub is in the house while the client is in a detached garage. We have a gate at the garage and it is controlled via the garage hub. In addition to having 2 HE hubs, I also have 2 Lutron Pro bridges. I want to be able to open the gate via a Pico remote while in the house. I added the Pico to the Lutron house account (there are 2 Lutron accounts - 1 for the house, the other for the garage). I programmed button 3 to open the gate but it's not working. Should this Pico remote be added to the house Lutron account or the garage Lutron account? Likewise, should I program the Pico via the house hub or the garage hub? Although I have other Pico remotes working well within the system, this is the first time I have attempted to have a Pico remote in the house control a device in the garage.
You don't need to choose which hub to add the Pico to; you can add the Lutron integrator app to both and just choose to add the devices you want. That being said, it should also work as a button device via HubConnect.
Since your gate isn't working with the Pico and the device it's controlling is on the other hub, there are two possible points for failure. One is the automation you've used to configure the Pico (I'd probably use Rule Machine with a "Button Device" trigger type, which exposes the old Button Controller UI, the dedicated app for which you could also use if you have it installed still/already). The other is the gate device via HubConnect. You can take some steps to troubleshoot which: does the gate device on the "remote" hub work via HubConnect? If so, the problem is likely your automation, and you may need to share more details to assist with troubleshooting (e.g., how you have it set up).
The gate can be opened via a Pico remote I have in the garage as well as via the Dashboard in Hubitat app. Both work great. So I know it works. The garage Pico opens the gate via a Simple Lighting rule. I set up the same rule for the new Pico that will remain in the house and it does nothing.
Additional Info: I am able to open the gate while in the new Pico device in Hubitat.
This makes me think that the problem likely resides in the rule. Would that make sense?
Hubitat supports multiple Lutron SmartBridge PRO integrations. Simply add them both to both of your Hubs.
In this manner, any Pico anywhere will be available to either hub. You can confirm that via Live Logs on the Garage Hub and pressing a Pico inside the house.
You'll see something like this:
dev:150 2019-11-19 10:50:03.285 am info rcvd: DEVICE,2,4,4
dev:150 2019-11-19 10:50:03.247 am info rcvd: DEVICE,2,4,3
dev:150 2019-11-19 10:50:01.334 am info rcvd: DEVICE,2,2,4
dev:150 2019-11-19 10:50:01.250 am info rcvd: DEVICE,2,2,3
the same event, on a Hub that uses that Pico shows:
2019-11-19 10:51:37.875 am info Office Pico button 5 was pushed
dev:9 2019-11-19 10:51:37.847 am info rcvd: DEVICE,2,4,4
dev:9 2019-11-19 10:51:37.695 am info rcvd: DEVICE,2,4,3
dev:1 2019-11-19 10:51:36.342 am info Office Pico button 1 was pushed
dev:9 2019-11-19 10:51:36.188 am info rcvd: DEVICE,2,2,4
dev:9 2019-11-19 10:51:36.033 am info rcvd: DEVICE,2,2,3
For me, I have one SmartBridgePro, but all my hubs have the Lutron Integration. Thus I CAN have each hub respond to a Pico independently... I don't need that, haven't done it, but I can see that I could 
You just need the same as I have, doubled up.
I don't see "slow" for my Pico's either... more specifically, yes, I see that 1'st Lutron connection after a reboot, but after that, it's as fast as I could want. And remember, my Pico expectations were formed by Staples Connect which also had a ClearConnect Radio internally, like Wink.
I just re-tested the two Lutron Hub scenario that I (thought I) recalled. Both scenarios work. Pico and controlled device on same Lutron Hub and Pico and controlled device on separate Lutron Hubs. I am going to delete my earlier comment to avoid any potential confusion.
The Pico has been added to each Lutron bridge and both HE hubs have a Simple Lighting Rule that should open the gate. The rule will not work from Pico when I am in the house. It works fine when I take the Pico with me into the garage.
If I understand it correctly, the consensus is that it should work from either. Do I have that right?
Someone with two Lutrons will have to answer if one Pico (aka "The Pico") can be paired to both. The Live logs should have all the answers.
I've been imagining a Pico in the Home, and another in the Garage so that wherever you are, a Pico is nearby, no 'carry' required. Maybe I didn't read the original query correctly.
I only carried the Pico to the garage to test it there. My plan is to leave it in the house.
My test today was a Pico on one Lutron Hub, a small RM 4 rule (button 1 press turn light on, button 5 press turn light off), and a Lutron Dimmer on another Lutron Hub. That worked. I don't have a second Hubitat to add to the test mix.
My two Lutron Hubs are close enough together (just separated by one floor) that range is not a problem and both devices are almost certainly communicating to the Lutron Hub on which they were installed.
When I moved from Caseta to RA2, I was shocked to see that adding it to RA2 (with the usual button-hold) did not remove it from Caseta. Both saw the events just fine. I have no idea what Lutron uses to associate these devices or if a "reset" on the Pico, assuming there is even a separate procedure for that, might have done so. I wouldn't put it past Lutron for it to not, but I can't say that it isn't reliable. 
(I also don't know if this would be true with two Casetas.)
I guess I was confused above, though. I thought you only had one Lutron Bridge. If the Pico responds to both--and I wouldn't be surprised--there's no reason this shouldn't work, just a matter of figuring out where the problem is. I'm not sure that's what you're really trying to do, though--it's just a device on a remote Hubitat that's the issue, right?
Update: As I had mentioned, I had created a Simple Lighting Rule on the house server hub to open the gate. It never worked. I then created a Simple Lighting Rule on the garage client hub. It worked fine as long as I took my Pico into the garage with me. No real surprise there.
In the last few minutes I created a RM rule on the house server hub. It didn't work but then it occurred to me that I still had the Simple Lighting Rule intact on that same hub using the same Pico and the same Pico button as the trigger. So I had unwanted redundancy. I removed the Simple Lighting Rule on the house server leaving only the new RM rule. And it works! I won't even try to explain it but it works.
