Hello,
The cooler on e-bay ought to do just fine.
I would not suggest locking your horse into her stall for safety reasons. Should the building have to be evacuated or should she have problems (like getting cast) in the stall, you and others would have quick access to assist her.
Since she has a long coat, after you bathe her, apply whatever coat shine you are going to use (ie: Showsheen, Pepi, etc.) to her while she is still wet, sweatscrape it through her coat, and then let her dry. You will get a more even result, as in the hair won't stick together and you won't get a streaky shine. You can apply more shine later before you show.
You only need the clear hoof polish. If the surfaces of her feet are really rough, it will be a good idea to blend them a little with sand paper. Scrub the white parts really well. I have used a neutral paste wax under the hoof polish. It will not change the color and helped to smooth the surfaces out, and enhanced the shine of the hoof polish. Apply it, let it dry and buff, then apply the polish over it.
If she is really fuzzy, you might only get away with using a #10 blade on her. You really don't want obvious layers. Do do her ears if you can. Clean up her lower legs, bridle path, throatlatch and between the jaw bones, her muzzle.....if you can blend well you may try to shorten some of the fuzz around the base of her ears and beside the bridle path.
Make sure that you, your horse and your tack are as spotless as possible - it will convey that you do care about your presentation and respect for the judge's opinion. Make sure your bit and stirrups do not get neglected.
Wash the mane and tail thoroughly - it will help tame the scuffies a bit.
Do braid.
You can edit this list, but would I take:
Shampoo, conditioner, sweat scraper, sponge, towels, cooler, spray shine (Important - if you use a silicone based shine like showsheen DO NOT apply it to the saddle area...your saddle WILL slide.), hoof polish, rubber bands, mane comb, alligator clips or the big clampy clips (for holding mane & tail sections), hair spray or setting gel (sometimes the gel comes in handy if you have little whisps that are sticking up that need to be schlocked down into place), clean brushes, hoof pick, sand paper, paste wax, hoof polish, more towels, baby oil (or alternate shiny substance for the face - if her facial hair is very long, use sparingly to keep hair from sticking together),
an easy to use tack cleaner for touch ups ( I use Bick 4 for between class wipes), and baby wipes. Tack (extra pad and extra reins.)
For yourself, comb or brush, mirror, whatever makeup you prefer, whatever method of restraining your hair that you prefer, oversized jumpsuit or sweats to wear over your breeches to keep clean, show clothes, hat/clothes brush, the previously mentioned Tide Stain Stick, a damp cloth for boots, spurs or whip if you use them.
Liniment, wraps, blankets or sheets. Lunge line, extra halter, extra lead, leather punch (optional), hammer, twine, chain or cable for tack stall, padlock, buckets, stall pick, bucket or tub for manure.
Plan on extra hay (I usually plan on 1/2 bale/day). Pre-package your horse's grain meals...zip-lock bags work great.
First aid items for you both - vet-wrap, ointments or wound sprays of your choice, band-aids, aspirin, sunscreen, etc.
Trash bag. Snacks and beverages for you. Pens, Copy of Registration papers and membership cards.
If Major posts her list then you can use both for ideas of what will work best for you.