How can I find tight asymptotic bounds for this half-history recurrence relation?
It is $n^{\Theta(\log n)}$, although I'm not sure exactly what the constant in the theta is. For the upper bound (the one you already have), note that, even without the $n^2$ term but with a base case of $1$ rather than $0$, this recurrence is dominated term-by-term by the recurrence
$$U(n)=nU(\frac{n}{2}).$$
For the lower bound, note that it dominates the recurrence
$$L(n)=\frac{n}{2}L(\frac{n}{4}).$$