A horse you are afraid of is not a horse you should own. You are afraid...deeply so of this animal.
This horse has issues and unless you want to once again risk injury to you or your daughter...no, just no.
What sounds like sudden onset of now bucking with a new leaser...
I would absolutely be looking for a tack problem as that kind of reaction is exactly what horses do when confronted with pain and the riders refusal to listen to the horses warnings.
Sell him, yes indeed.
Not to be nasty but...
I would be questioning that trainer and where their head is at if they know this horses behaviors and would put a 5 year old on him...yea, no.
16.2 hands of power is not
where you put a 5 year old...not on your horse with his known problems. Never! Sell the horse.
Disclose he is touchy about his hind end both from the ground and astride if touched...to me it isn't a deal breaker but something need aware of.
You need to get your horse evaluated for what he is worth.
If you can find a sale barn, put your horse in it knowing his worth and what you must pocket to find another horse in his place.
Sale barns usually have better than average riders who will handle bad behaviors astride, make corrections and have a larger amount of people looking for sale prospects come through the door.
A sale barn will also ride the horse at all gaits in demonstration shown...the sale barn staff will ride, part of why they get their commission price/paid and you don't have to nor do you need to be in attendance during potential sales/showing of the animal. Actually it is better if you, the owner is not
I can only imagine a large 16.2 hand Quarter Horses worth just because people want size, they want the bigger the better today.
You say he can be a "sweetie", but he has this quirk...
You've got the "big", now you need to find "the better" both for you and for your daughters safety. What size horse do you need?
where I would start...
Right now, dead of winter but spring coming along is a buyers time and market...
Take some lessons on any new horse prospect, handle the horse entirely on your own so you find any "quirks" like what you have and can't deal with if honest.
Don't buy a baby or "just-started" but more a horse with a bit of age so chances are the horse is set in their ways, habits are formed and what you see is what you get personality found.
You need to look not just for you but for your daughter to be safe with, around and astride so she can ride and enjoy special time with her mom.
Those horses are absolutely out there...
Today, look at the teenage aged horse as horses often ride into their late 20's and sometimes those with a bit of age on them can be had for dirt cheap and are often over-looked for the school-masters they are = worth their weight in gold.
When you find "your new-one" do a PPE to make sure health issues are not a issue.
Once this currently owned horse is found a new home, then go find yourself a steady riding partner.
Take some lessons in the meantime to get your confidence back. Sell, you bet...
In the meantime... Do not
allow your daughter on this horse regardless of what this trainer says...the horse is to big for her, has issues that could...don't even go there...just no to her being in close proximity to the animal except to pet the neck, even then only with strict supervision.
Your daughter is a "peanut" compared to this horse...most 5 year olds are tiny.
That "peanut" does not need to have its shell cracked by your kicker, bucker or anything else he is doing that could put her in jeopardy.
If she is wanting to ride...put her in lessons on size appropriate animals
of larger pony to smaller horse.
Just think of where her small legs would be on your current horse of this size...it isn't safe for her, period.
Sorry, but no, no way-no how would my child be anywhere near a horse I am afraid of to ride and handle and who has hurt me, the experienced rider now 2x...just no way!
Your guts instincts are right...
Sell this animal, and find another, a match you can enjoy riding, spending time with and owning.
A animal your daughter can also ride and share special times with her mom.
Look for and find a animal who has a very forgiving nature and accepts bumbles as nonchalantly, not as a explosion...
Oh...the perfect horse is out there waiting for you...
Sell this one and then enjoy the search for a new partner, one safe for both of you to be around and astride.
If this was my horse, the ad would be made, the sale barn and evaluation done and the horse headed for a new home... get moving.
It is a buyers market but also a sellers for I bet what you have sitting in a stall you are frightened to ride and handle.
You have a dull diamond just waiting for a experienced rider/handler...just needs a bit of polishing.