Can I express $\gamma$ as an analytic function of $a,b,c$, and $\alpha$?
So far, I have computing $p_1$ from $a,b,\alpha$, then solved numerically for $p_2$ by intersecting the ellipse with a circle with radius $c$, then solved analytically for $\gamma$ using $\alpha, a,b$ and $p_2$. I am looking for a better way that, ideally, does not require a numerical solver in the middle.
The intersection of circle and ellipse generates quartic equation. The quartic equation is solvable in algebraic functions, so the answer to your question is yes, there is an analytic solution. however the resulting formula is so big, that a Brent or even Newton method would work way faster.
To illustrate what we are talking about. Here is the answer to one of the coordinate of an equivalent problem of $$ \begin{cases} a x^2+bxy+cy^2=1,\\ (x-1)^2+y^2=1 \end{cases} $$ $$ x=\frac{b^2+2 c^2-2 a c}{2 \left(a^2-2 c a+b^2+c^2\right)}-\frac{1}{2} \sqrt{\frac{\left(b^2+2 c^2-2 a c\right)^2}{\left(a^2-2 c a+b^2+c^2\right)^2}+\frac{\sqrt[3]{128 c^6-384 c^5+384 a c^4+384 c^4-288 b^2 c^3-768 a c^3-128 c^3+384 a^2 c^2+432 b^2 c^2+384 a c^2-384 a^2 c-288 a b^2 c-144 b^2 c+108 b^4+128 a^3+144 a b^2+\sqrt{11664 b^8-6912 c^3 b^6+10368 c^2 b^6+31104 a b^6-62208 a c b^6+10368 c b^6-6912 b^6+27648 a c^4 b^4-6912 c^4 b^4+27648 a^3 b^4-55296 a c^3 b^4+13824 c^3 b^4-6912 a^2 b^4+55296 a^2 c^2 b^4+13824 a c^2 b^4-6912 c^2 b^4-55296 a^2 c b^4+13824 a c b^4}}}{3 \sqrt[3]{2} \left(a^2-2 c a+b^2+c^2\right)}+\frac{4 \sqrt[3]{2} \left(4 c^4-8 c^3+8 a c^2+4 c^2-6 b^2 c-8 a c+4 a^2+3 b^2\right)}{3 \left(a^2-2 c a+b^2+c^2\right) \sqrt[3]{128 c^6-384 c^5+384 a c^4+384 c^4-288 b^2 c^3-768 a c^3-128 c^3+384 a^2 c^2+432 b^2 c^2+384 a c^2-384 a^2 c-288 a b^2 c-144 b^2 c+108 b^4+128 a^3+144 a b^2+\sqrt{11664 b^8-6912 c^3 b^6+10368 c^2 b^6+31104 a b^6-62208 a c b^6+10368 c b^6-6912 b^6+27648 a c^4 b^4-6912 c^4 b^4+27648 a^3 b^4-55296 a c^3 b^4+13824 c^3 b^4-6912 a^2 b^4+55296 a^2 c^2 b^4+13824 a c^2 b^4-6912 c^2 b^4-55296 a^2 c b^4+13824 a c b^4}}}+\frac{4 \left(-2 c^2-c+a\right)}{3 \left(a^2-2 c a+b^2+c^2\right)}}-\frac{1}{2} \sqrt{\frac{2 \left(b^2+2 c^2-2 a c\right)^2}{\left(a^2-2 c a+b^2+c^2\right)^2}-\frac{\sqrt[3]{128 c^6-384 c^5+384 a c^4+384 c^4-288 b^2 c^3-768 a c^3-128 c^3+384 a^2 c^2+432 b^2 c^2+384 a c^2-384 a^2 c-288 a b^2 c-144 b^2 c+108 b^4+128 a^3+144 a b^2+\sqrt{11664 b^8-6912 c^3 b^6+10368 c^2 b^6+31104 a b^6-62208 a c b^6+10368 c b^6-6912 b^6+27648 a c^4 b^4-6912 c^4 b^4+27648 a^3 b^4-55296 a c^3 b^4+13824 c^3 b^4-6912 a^2 b^4+55296 a^2 c^2 b^4+13824 a c^2 b^4-6912 c^2 b^4-55296 a^2 c b^4+13824 a c b^4}}}{3 \sqrt[3]{2} \left(a^2-2 c a+b^2+c^2\right)}-\frac{4 \sqrt[3]{2} \left(4 c^4-8 c^3+8 a c^2+4 c^2-6 b^2 c-8 a c+4 a^2+3 b^2\right)}{3 \left(a^2-2 c a+b^2+c^2\right) \sqrt[3]{128 c^6-384 c^5+384 a c^4+384 c^4-288 b^2 c^3-768 a c^3-128 c^3+384 a^2 c^2+432 b^2 c^2+384 a c^2-384 a^2 c-288 a b^2 c-144 b^2 c+108 b^4+128 a^3+144 a b^2+\sqrt{11664 b^8-6912 c^3 b^6+10368 c^2 b^6+31104 a b^6-62208 a c b^6+10368 c b^6-6912 b^6+27648 a c^4 b^4-6912 c^4 b^4+27648 a^3 b^4-55296 a c^3 b^4+13824 c^3 b^4-6912 a^2 b^4+55296 a^2 c^2 b^4+13824 a c^2 b^4-6912 c^2 b^4-55296 a^2 c b^4+13824 a c b^4}}}-\frac{\frac{8 \left(b^2+2 c^2-2 a c\right)^3}{\left(a^2-2 c a+b^2+c^2\right)^3}+\frac{16 \left(-2 c^2-c+a\right) \left(b^2+2 c^2-2 a c\right)}{\left(a^2-2 c a+b^2+c^2\right)^2}+\frac{32 c}{a^2-2 c a+b^2+c^2}}{4 \sqrt{\frac{\left(b^2+2 c^2-2 a c\right)^2}{\left(a^2-2 c a+b^2+c^2\right)^2}+\frac{\sqrt[3]{128 c^6-384 c^5+384 a c^4+384 c^4-288 b^2 c^3-768 a c^3-128 c^3+384 a^2 c^2+432 b^2 c^2+384 a c^2-384 a^2 c-288 a b^2 c-144 b^2 c+108 b^4+128 a^3+144 a b^2+\sqrt{11664 b^8-6912 c^3 b^6+10368 c^2 b^6+31104 a b^6-62208 a c b^6+10368 c b^6-6912 b^6+27648 a c^4 b^4-6912 c^4 b^4+27648 a^3 b^4-55296 a c^3 b^4+13824 c^3 b^4-6912 a^2 b^4+55296 a^2 c^2 b^4+13824 a c^2 b^4-6912 c^2 b^4-55296 a^2 c b^4+13824 a c b^4}}}{3 \sqrt[3]{2} \left(a^2-2 c a+b^2+c^2\right)}+\frac{4 \sqrt[3]{2} \left(4 c^4-8 c^3+8 a c^2+4 c^2-6 b^2 c-8 a c+4 a^2+3 b^2\right)}{3 \left(a^2-2 c a+b^2+c^2\right) \sqrt[3]{128 c^6-384 c^5+384 a c^4+384 c^4-288 b^2 c^3-768 a c^3-128 c^3+384 a^2 c^2+432 b^2 c^2+384 a c^2-384 a^2 c-288 a b^2 c-144 b^2 c+108 b^4+128 a^3+144 a b^2+\sqrt{11664 b^8-6912 c^3 b^6+10368 c^2 b^6+31104 a b^6-62208 a c b^6+10368 c b^6-6912 b^6+27648 a c^4 b^4-6912 c^4 b^4+27648 a^3 b^4-55296 a c^3 b^4+13824 c^3 b^4-6912 a^2 b^4+55296 a^2 c^2 b^4+13824 a c^2 b^4-6912 c^2 b^4-55296 a^2 c b^4+13824 a c b^4}}}+\frac{4 \left(-2 c^2-c+a\right)}{3 \left(a^2-2 c a+b^2+c^2\right)}}}+\frac{8 \left(-2 c^2-c+a\right)}{3 \left(a^2-2 c a+b^2+c^2\right)}} $$