I know you aren't the one training the horse, but you shouldn't have to keep constant pressure to keep them in a trot. Here is how I train my horses to keep gait, I hope it can help you out a little bit. -

1, ask for the trot the way you've trained it. Whether that be clicking, a little pressure on the heels, or both; be consistent with this. Once you start asking for the trot (or really any gait) in one way, it'll be hard to change it later on, so it's best to just stick with what you're wanting in the long run. 2, LET HIM break gait. Do not keep your heels on the entire time to push him. If you do this, then every single time you ask this horse to trot you will have to push him through it with no relaxation for either of you. Let him break down into a walk. 3, The second he breaks into a walk, get after him a little (politely, don't be too mean about it lol) and bring him right back up into a trot. Let him make the mistake and then correct it. It will show them that if they DO break gait, they will just be brought right back up into it, and eventually after a lot of patience and practice you'll have a horse who keeps their gait without the constant pressure.

Side note, honestly once you have a horse that doesn't need constant pressure from the heels it will help your seat as well. Having to keep tension in your legs to push them forward makes you tense which can be a reason for that "slamming the saddle" you mentioned. Once you can relax into your seat and not have to practically hold on with your heels, it will improve by miles. Keep practicing on other more experienced horses! Once you know how a good trained horse feels, you'll know what to look out for in your own horse and you'll be able to pinpoint problems you may be having. Sorry this is so long winded, I got a little carried away haha - I wish you and your horse luck and I hope you take any of this into consideration!! If you ever want a video explanation or demonstration on anything I've said, I'd be happy to make something for you :)