Honestly, I’d probably pay the money to get it professionally cleaned… they’ll do an awesome job for you, and eliminate all the effort you are going to put into this AND THE STRESS
That said, my Wedding Dress this time round was a fairly inexpensive gown suitable for a Beach / Destination Wedding (which is what we did for our Elopement).
It was all synthetic, and not pouffy… so it was easy to put it into the Washing Machine on the gentle cycle (and small enough to fit into an Extra Large Lingerie Bag). It turned out really well… even the hem came clean (but it wasn’t uber dirty to begin with… no mud, trodden in mess etc).
It was also Ivory… so I just used regular Laundry Detergent (HE Tide Vivid White Bleach… which is safe for colours) … I might have washed it twice (back to back cycles), cannot really recall. But it turned out perfectly.
NOW if it was WHITE and DIRTY like yours is, and I was determined to do it on my own, I’d look into a product called DiDi 7, that you can buy off the Internet. I’ve used this product for years on WHITES (and ONLY WHITES) that need to get the hard stuff out of… wine, blood, grass stains, mud, grease, oil, make-up etc.
As your dress is completely synthetic there should be no fabric issues (altho I don’t know about the netting / tulle issue)… in regards to shrinkage, or reaction to cleaning products (for the most part). That is unless you have applied lace on the dress, or beads, crystals or other decorations… cause a lot of times those items are “glued on” with modern dresses, not sewn on… and they tend to come off if the gown is laundered vs dry-cleaned.
If your dress has any embellishments, or you are in the least bit unsure of the fabric, etc.
I’d say skip it, and just pay someone to do it for you professionally.
Hope this helps (somewhat)
EDIT TO ADD – You have to remember too that if you screw up there is no recourse, if the Dry-Cleaners do you usually will get some financial compensation. And that can make a world of difference too IMO