And keeping vibration down.

Any ideas?  This generator will 'permanently' stay in the back of the truck.

Give us some idea of what else you will use the bed for? Do you want to enclose it and protect it from the weather/thieves? Or leave it open?

I assume you will mount it at the front of the bed, so it doesn't interfere with hooking up  your Gooseneck ( if you have one).  I'd probably place it next to one side or the other, So I could fuel it with climbing in the bed.  So for me it would be on the same side of the truck as the fuel door for the truck.  I'd also raise it off the truck bed high enough to make draining the oil out of easy.  So generators will drain the oil while sitting flat. Others need to be tipped in order to get the old oil out. So your mounts MAY need to be easy to remove every time you change the oil.

I would probably mount it on a piece of heavy plywood or steel plate that was mounted on some rubber grommets or maybe even springs.  You might take an floor mat and cut it to shape an place a couple layers of it under the plywood.

There are a number of ways to do it, depending on your skills and what material you access to. Of course a steel plate would be more durable, Plywood would have to be replaced periodically.

I don't use the truck bed as a truck bed, just for the trailer (gooseneck)
It's currently next to the cab on the passenger side so I don't have to get in the bed at all to hook-up or start.

I'd like to *permanently* bolt it down (take it out for servicing) as it will probably never need to be taken out otherwise.

I'd also like to leave it as low-profile as possible.  I'm afraid to put it in a box because, in Texas "surface of the sun" - it was 95* yesterday, Easter!  How's that for spring weather.  ugh.