Should I Buy The Dealer's Hitch

If you're willing to pay the $1000's extra for an F250 over an F150, then why are you quibbling over the $300 of shop time to have the hitch installed? That's 2 hours of shop time these days, which is about right.

Then again, you 100% will need to adjust it yourself once you get home and get the trailer loaded. You're only paying for them to assemble the hitch and physically put it on the truck/trailer.

Guess I'm not really helping.

For a 9000GVW trailer, go with the F250. Even with modern trucks I still maintain that 7500 loaded is the COMFORT limit for an F150.

I've never towed with an Equal-i-zer hitch, I'm a Reese dual-cam guy (had several of them due to the heavy tongue weight of my Reflections). I think Equal-i-zer is the most popular since that what dealers push. However, they are very noisy, my traveling companion has one on his Airstream/Mercedes. I can tell he's arrived as soon as he pulls into the campground. That's his number 1 complaint with it and it's hard to snap-up (yes, he does use the tongue jack as well). The only thing to caution you about is: height of hitch receiver on a F250. You'll probably need a long drop shank.
Having towed with several F150s (hundred of thousands miles), it's a better truck than the Super Duty. I have had 3 Super Duty's. My current one is a F350/PS 6.7 Lariat and I tow 42' fifth-wheel and it's one hell of a good truck, but the last F150 was a better truck all around (except capacity & power). Keep in mind, that there are F150s and then their are F150s. You need to buy a truck properly equipped to tow the load you have.
As far as having the dealer supply and install the hitch... There are a lot of things you don't know about setting it up properly and it will make all the difference in your towing experience.

We have an Equilizer that I installed myself, it isn't that hard if you have basic tools and some mechanical ability.

You didn't say the weight and length of your trailer, but in most, if not all cases, weight distribution and anti-sway bar are recommended. Some may say otherwise with the anti-sway stating that a properly loaded trailer will eliminate the need for it, but we have always had them for our bumper pulls with no regret.