Your time deadline limits you, but when you return...this below is important to do..
Be safe and better to be late attending to some much needed maintenance and having correct rated ball for the trailer.
My impression from the picture shared is you have not greased your hitch in some time, nor added spray lube to the working latch of the hitch either...
So, some maintenance is needed now and continually as you use the trailer to have a snug, good and safe fit hitching trailer to tow vehicle.
Use a marine grease to lube the ball so it glides not squeak or rub when towing the trailer...
There are basics of trailer hitch components many do not know...and should.
A trailer hitch ball is not
one size fits all...
They are rated, they are several different materials used and your trailer will tell you what it needs
on a plate that should be someplace on that trailer...
I found you a article about hitches, the different styles and ratings and why it is important to have the correct matched equipment to job needing done is so important. https://www.etrailer.com/faq-hitch-ball.aspx
Bottom line is if you don't match your components you run a very good risk of over-taxing the equipment, it breaks and you have devastation in animals hurt, people hurt and potential of death and you are
The swing arm part of the trailer hitch is not
open fully so no, you won't fit the appropriate sized ball in the receptacle.
Lithium grease applied so that hinge moves easily is important, watch the fingers they can easily pinch you when it gives and opens..
Grease every part of that hitch as it is very dry looking and that is just not good for the trailer as it encourages rust to form...
Do check the hitch plate for what rated ball you need...I can see it stamped on the hitch non-moving section in your picture but can't read it all..
Do look at what your slide in section needs, whether a regular fit or a tall fit ball is needed...
Do torque the ball closure washer and nut at specific pounds so it is attached but not damaged by over-tightening.
Do grease either the ball or the inside of the hitch, makes no difference cause when you hook together it gets on both, but it will make a difference in towing when coupled.
Do also grease the part of the hitch that retracts the hitch wheel and if wheel detaches, grease that part too so easier to remove.
I would also say if your hitch appears as dry of grease as it is, the rest of your trailer is possibly also.
Every door hinge, window, latch closure need some attention, done as often as needed and at least once a year minimum...it is the life of your trailer, period.
When not using the hitch if you remove it from the tow vehicle cover the ball with a plastic bag so reducing dirt/sand and grit from getting into the grease if you leave on the ground as your picture currently shows...we also cover the trailer hitch part itself so our dogs don't wear grease if they rub against the hitch...yuck!
Remember to have someone qualified and knowing what they are doing check and grease wheel bearings, check brakes and hubs for leaking cylinders and to grease fittings if the trailer has them.
Yearly maintenance is important and cheap to do to catch small issues before they become large issues.
A bit of preventative maintenance is so very important when doing horses as road-side breakdowns are dangerous when loaded especially.
Your trailer also has safety "breakaway" brakes by appearance of the thin cord on the hitch....
Make sure it is working, battery is charged if yours works that way and you thread the cord correctly through the tow chains that should a hitch failure occur your horses have the best chance of survival cause the trailer isn't a runaway but grinds to a safe, straight stop.
But yes, standard ball for horse BP is 2 5/16...depending on weight of trailer depends on ball selected and height range needed to settle the trailer correctly on the ball and have the locking mechanism engage on it.
If you are unsure, take the trailer to someone knowledgeable and have it checked for being right...it is ultimately your life and that of your animals towed depending upon your actions done.