Many thanks. I could not carry all groups of order 512, but my colleague Maria Guedri was running a program under GAP to do it. She has ruled out about 1/3 of such groups or more.

@Tilman: Only a remark not related to the topic: The identity $$\dim (U + V) = \dim U + \dim V - \dim (U \cap V)$$ is valid only for finite dimensional spaces, but if one writes it as follows $$\dim (U + V) + \dim (U \cap V)= \dim U + \dim V$$ it is valid for all vector spaces.

The following paper may be related to the question: Neumann, Bernhard Hermann. Commutator Laws in Algebraic Systems. Centre for Mathematics and its Applications, School of Mathematical Sciences, Australian National University, 1996.

If $p=2$, then $[F,F]/[F^2,F]$ is generated by the image of $[x,y]$ in the quotient group. So the quotient is cyclic if $p=2$. Why you think that the exponent must be finite in the case $p=2$? $x$ and $y$ are free generators of $F$.

@GeoffRobinson. Actually it is equivalent to ask if there is an infinite finitely generated group with exactly 3 maximal subgroups (without any other restriction on the indices).

@AliTaghavi. I will shortly write the mates. Those are of length 14.

@HJRW. I mean the quotient by the Frattini subgroup.

Is it possible to give an explicit presentation for the group of order $p^7$ in your last EDIT? I have used the same relations as the Lie algebra with the relations of 7th powers of all generators. This gives me a group of nilpotency class 5, expoenent 7 and order $7^6$ whose automorphism group is of order 242121642. Is this because my group is not obtained from BCH?

@YCor. There is no example of order $3^7$.