I think that the problem is that with your \$\beta\$ the \$K_p\$ term \$(R_1/R_2(1+C_2/C_1)-R_f/R_g)\$ goes negative, so you haven't sufficient design constraints to get \$K_p = -0.5\$.

Also, your equations don't match your schematic - the capacitor values are swapped.

The problem you are trying to solve is to find values for \$\alpha\$ and \$\beta\$ that satisfy the two equations for \$K_P\$ and \$Q\$.

The problem you have solved is to find values for \$\alpha\$ and \$\beta\$ that satisfy the equation for \$\frac{|K_P|}{Q}\$, but not the separate equations for \$K_P\$ and \$Q\$. You need to add one of these in as a requirement.

I suggest you write these two equations out, eliminate one variable (\$\alpha\$ or \$\beta\$) and solve. If I have done it correctly, I think you will find a separate condition \$|K_P| > \sqrt{8} Q - 1\$, which your attempted design fails.

Also, your equations don't match your schematic - the capacitor values are swapped. This doesn't really matter as you assume they are equal, but it's still wrong.