Plot Hole?

I don't think you understand the difference between foster and adoption. This wasn't a court appointed foster family, they clearly stated in the film. This is mutually agreed upon family from both and Evelyn and Frank. First of all when the family denies him his visitation rights to hide the fact that his mother is living in their guest house and not to mention they got rid of the cat they promised to keep, they have broken the law and a verbal contract. Secondly, The foster parents are only entitled to the child if both Frank and Evelyn mutually agree. It's clear that the only reason Evelyn wants Mary to dedicate her life to mathematics is to solve the proof that her daughter was working on her entire life, so once Evelyn has the solution to the mathematical proof she no longer needs Mary and agrees that Frank should be granted his rights as guardian.

Well, it was supposed to be a mutual compromise, that the girl won't live with either the uncle or gandma, but in foster care. Which is close to the uncle's place and they both have some visiting rights. As shown in the movie, when Chris comes with some toys to meet her.

Now, since the Grandma broke the law/rules, by coming over to the foster house (presumable bribing the foster parent(s), as you can see the wife doesn't seem to like this arrangement and tells Chris where they are).

This gives Chris evans' character more of an advantage if he now went to court and showed that the grandma broke the law, yada yada yada. So, now he just went and brought the girl home and will deal with the law part of it later.

However, it would never come to it, as the Grandma will just withdraw all her petitions as she got what she wanted.