For this and other Macaulay2-related questions, I highly recommend the Macaulay2 google group.
In general, there are some "exploratory" techniques (i.e. not quite a proof) which are more efficient and may indicate whether your original approach will yield the desired outcome. For your example, I think it makes sense for starters to work with coefficients over a large prime field and plug in random constants for $m$ and $t.$
In the example below, I take $I$ to be the ideal $\langle f_1, \ldots , f_{12}, f_{13} \rangle $ in the polynomial ring $\mathbb{F}_{10007}[a,b,c,d,u,v,w,r,p,q,z,s],$ where $f_{13}$ is a random inhomogeneous polynomial of degree $1$ (this can be interpreted as random chart on the projective space $\mathbb{P}^{11}.$) The computation that $\dim I = 3$ takes about 1s on my laptop. This suggests (but doesn't quite prove) that $V_{\mathbb{C} \, \mathbb{P}^{11}} (f_1, \ldots , f_{12}) \ne \emptyset .$ Your conjecture may still be correct---as you mention, this is only a subsystem of the one you've derived...
pr = 10007
FF = ZZ/pr
t = random FF
m = random FF
R = FF[a,b,c,d,u,v,w,r,p,q,z,s]
f1 = u*p+v*z-t*(a*u+b*w)
f2 = u*s+v*q-t*(a*v+b*r)
f3 = w*p+r*z-t*(c*u+d*w)
f4 = w*s+r*q-t*(c*v+d*r)
f5 = v*p+m*u*z-t*(b*u+m*a*w)
f6 = v*s+m*u*q-t*(b*v-m*a*r)
f7 = r*p+m*w*z-t*(d*u+m*c*w)
f8 = t*s+m*w*q-t*(d*v+m*c*r)
f9 = u*u+v*w-a*p-b*z
f10 = u*v+v*r-a*s-b*q
f11 = w*u+r*w-c*p-d*z
f12 = w*v+r^2-c*s-d*q
f13 = random(1,R)-1
G = groebnerBasis(ideal(f1,f2,f3,f4,f5,f6,f7,f8,f9,f10,f11,f12,f13), Strategy=>"F4");
dim ideal leadTerm G