How precise are BQPSPACE measurements?

This is in a similar spirit to another question I asked here.

Let’s say I am given a $k$-local Hamiltonian $H$. We know that $||H|| leq 1$. Let the ground state be $|psi_{0}rangle$, with energy $E_{0}$. Let $U$ be an $n$-qubit unitary such that
begin{equation}
U |psi_{0}rangle |00cdots0rangle = |psi_{0}rangle |E_{0}rangle.
end{equation}

Let $V$ be an $n$-qubit unitary such that
begin{equation}
V |00cdots0rangle = |psi_{0}rangle,
end{equation}

Can we implement both $U$ and $V$ in polynomial space (ie, with polynomially many qubits), with exponentially many gates from a universal gate set, upto an inverse exponential (in the number of qubits) precision? In other words, let $U’$ and $V’$ be what we can actually implement with exponentially many gates, and let $|psi_{0}’rangle$ and $E_{0}’$ be the ground state and the ground energy we get by applying $V’$ and $U’$ respectively. What is the relation between $|psi_{0}rangle$ and $|psi_{0}’rangle$, and $E_{0}$ and $E_{0}’$? Are they also within an inverse exponential additive error to each other?

If so, is this an alternative proof that the $k$-local Hamiltonian problem, which is complete for QMA, can be solved in BQPSPACE (the quantum analogue of PSPACE)?

Is this also why we do not know if QMA is contained in BQP? In other words, we do not know if there is a description of $U$ and $V$ which requires only polynomially many gates.