I lost my licence a few years ago because of "a vertical squint".It was picked up by my AME, initially and I was referred to the CAA's visiting eye specialist at LGW.
To try to answer your questions as best I can based on a lot of research over a harrowing few months....
If you have Hypophoria outside the limits you will not be granted a medical. IIRC the limit for an initial is 1 dioptre, and the CAA have applied for an allowance to let you have 2 dioptres if you are experienced and are renewing your licence.
The CAA view (no pun intended) vertical squints as likely to cause double vision, and I very much doubt you will be able to prove to them that it will not afffect your flying ability.
Here is the good news. If the cause of your verical squint is caused by paresis (weakening) of the main eye muscle than subject to a number of tests and an operation you should be able to get your eye balance back to 1 dioptre.
In my case there are no exercises that would have helped.
Carrots won't help!
The FAA can grant you a medical with a SODA limitation (SODA=statement of demonstrated ability). I am not sure whether they will grant his to an initial application but they will (at their discretion) grant it to experienced people.
Yes there is hope!
I flew for nearly 8 years suspecting that there was something wrong with my eyes, but the human brain is a marvellous thing and it just copes. I never once suffered from from diplopia (double vision) I am a professional pilot and was passing medicals and LPC/OPC's with no problems. Throughout my ordeal the CAA were marvellous, and contrary to some reports thay really do want you to keep flying. My local Opthalmic Surgeon did the opeartion and about 8 weeks later I was judged to be within limits and happily have my licence back.
I am apologise for not posting earlier but I don't often visit this forum. If you want more information ask away and I will dig out my notes.
Good luck