What are the possible alternative solvers available to handle these kind of problems below?
Gurobi does not support semidefinite cones larger than $2 \times 2$. You must use another solver for this problem.
I am using semidefinite programming with CVX Gurobi and my semidefinite constraint is as follows,
$$H'AH-Q \succeq 0$$
where $H'AH$ is a positive semidefinite matrix and $Q$ is a binary diagonal matrix.
Mosek can handle mixed-integer conic optimization problems.
https://docs.mosek.com/9.0/cmdtools/mip-optimizer.html
Another possibility is YALMIP:
https://yalmip.github.io/tutorial/integerprogramming/