I see you lucked out OP with it not raining during your show day, and it is nice you made the choice to scratch to prevent any injuries to your horse.
I am going to add what i do on rainy days, just in case anyone finds this thread and wants tips.
Bring everything. I am well known for having multiples of everything, even on a normal day and have even had other competitors come to my trailer to ask to borrow things.
I hate wet breeches, so on showery days i tend to bring every pair i own and swap when it stops raining, if it is a constant rain, then i just suck it up. The extras also help if you fall so you’re not putting muddy breeches into your saddle. Do not forget extra gloves, nothing is worse then putting on wet gloves.
If you show jumpers, it is nice to be able to swap out of water logged polos or boots.
I usually have a long break in between my classes, but it is never long enough for my pad to dry so i will bring extra and replace each time i tack up.
I have purchased waterproof quarter sheets for these rainy days. I will also bring multiple rain sheets to cover my horse. At the local shows, there is no stabling, so your just waiting around at your trailer. My trailer does not have an awning, so i bring a shelter (like a gazebo just without sides) that i have trained my horse to stand under, it helps alot on rainy and hot sunny days.
Practice. If you are willing to show in the rain, you need to practice in the rain. Your horse needs to be used to running into the driving rain, and stomping through puddles, as well as seeing many umbrellas.
After riding in the rain, i like to give my saddle a really good cleaning and finish off with neats foot oil.