What do it yourselfer skills do you have? You would have to check local code of which I will follow but never get permits.
You should be able to use Pex to run gas line to the fire pit. You can buy it in rolls so it will be one continuous pipe with just a connection on each end. No screwing galvanized pipe together or sweating copper joints. Long runs end up quick work. A walk behind trencher which many will call a ditch witch rented for a day will cost $300. Add the Pex and fittings DIY your looking at $500 total.
Heck, depending on distance you might hire out the trench job for $300 or less. Now a little old and beat up to want to wrestle a trencher I paid a neighbor just $200 recently for a two foot deep 40 yard long trench to a well for me to replace the plumbing and electric lines. Damn, Bidenomics it was a dozen years ago but I last rented a trencher it was $125 a day. Just my opinion, I am thinking from the perspective not changing propane tanks or having a large stationary one put in. I think they don't have curb appeal. If you don't want to run the gas line yourself I would
just get a quote from a plumber specifying Pex, it could be cheaper than you think.
I actually never heard of the word pergola, I had to look it up. I would just call that a patio cover. Not sure why you can't have a fire pit under it. How
many people BBQ under patio covers. I am sure treated lumber and maybe add some metal flashing should be fine.
