Analytic solution to definite integral problem

Some insights...

We can look at your integral as a Mellin convolution.

The Mellin transform is defined as

$$ \mathcal{M}\left\{f\right\}(s) = \mathcal{M}_f(s) = \int_0^\infty f(x)\,x^{s-1}dx$$

and the Mellin convolution of two functions is defined as

$$ (f *_{\tiny \mathcal{M}} g)(s) = \int_0^{\infty} f(x)\,g\!\left(\frac{s}{x}\right)\frac{dx}{x}. $$

In your case, we have $f(x) = (1-x)^a \theta(1-x)$, $g(x) = e^{-x}$, and we want to compute $(f *_{\tiny \mathcal{M}} g)(b)$.

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A nice property of Mellin convolution is

$$ (f *_{\tiny \mathcal{M}} g)(s) = \mathcal{M}^{-1} \left\{\mathcal{M}_f(s) \mathcal{M}_g(s) \right\}(s), $$

and using the table here, we have

$$ \mathcal{M}_f(s)\mathcal{M}_g(s) = \frac{\Gamma(a+1)\Gamma^2(s)}{\Gamma(s+a+1)}, \quad \text{for} \;\; \text{Re}(a) > -1, \; \text{Re}(s) > 0. $$

Now by definitions of the inverse Mellin transform and the Meijer G function, we get

$$ (f *_{\tiny \mathcal{M}} g)(b) = \Gamma (a+1) G_{1,2}^{2,0}\left(b\left| \begin{array}{c} a+1 \\ 0,0 \\ \end{array} \right.\right).$$

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We can apply the second identity here to simplify the Meijer G and express your integral as

$$ \boxed{\int_0^1 dx \; f(a, b, x) = e^{-b} \Gamma (a+1) U(a+1,1,b),} $$

where $U$ is the confluent hypergeometric function

$$ U(\alpha, \beta, z) = \frac{1}{\Gamma(\alpha)} \int_0^{\infty } e^{-zt}t^{\alpha-1} (t+1)^{\beta-\alpha-1} dt. $$

Other than identities here for specific $a$ and $b$, I'm not aware of a way to whittle down $U(a+1,1,b)$ to anything nicer.

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Edit

Upon looking at the definition of $U(\alpha, \beta, z)$, I realize one can make the substitution $x \mapsto 1/(t+1)$ to get

\begin{align} \int_0^1 (1-x)^a \exp\left(-\frac{b}{x}\right) \frac{dx}{x} &= \int_0^\infty e^{-b(t+1)} t^a (t+1)^{-a-1} dt\\ &= e^{-b^{\phantom{1}^\phantom{1}}} \!\!\! \Gamma (a+1) U(a+1,1,b), \end{align}

which is a much less roundabout solution.

I don't believe there is going to be an analytic solution for solving the integral for a. (I'm happy if someone proves me wrong).

So we need to do something practical. Here is some consideration.

You want to solve for $a_2$:

$$\int_0^1 dx\ f(a_2,b_2,x) - \int_0^1 dx\ f(a_1,b_1,x) = 0$$

Let's rename $- \int_0^1 dx\ f(a_1,b_1,x) = G_{-1}$.

With some initial good starting value, replace $a_2$ with $a_2 + d$ where the difference $d$ will be small. So you need

$$ G(d) = G_{-1} + \int_0^1 dx\ f(a_2+d,b_2,x) = 0$$

Now $f(a_2+d,b_2,x) = f(a_2,b_2,x) (1-x)^d $ and you can expand $(1-x)^d$ into a power series in $d$. This works, since you expand, in the integrand,

$$ (1-x)^d = \sum_{n=0}^\infty (\ln(1-x) )^n d^n / n! $$

EDIT: there was a wrong extra factor $(1-x)^d$ in the summands which was removed here and later.

and the terms do not diverge in the interval $x \in (0, 1)$ and the corresponding integrals exist as well. So you have

$$ G(d) = G_{-1} + \sum_{n=0}^\infty d^n \quad 1/ n! \int_0^1 dx\ f(a_2,b_2,x) (\ln(1-x) )^n = 0$$

Exemplarily, I chose some $a_2,b_2$ and I let Wolfram Alpha compute the first $n=10$ many terms $1/ n! \int_0^1 \cdots$ and one sees their values decrease rapidly with $n$.

So this leaves you with choosing some $N$ (depending on the desired accuracy), computing the integrals and then solving, for $d$, a polynomial equation $$ G(d) = G_{-1} + \sum_{n=0}^N d^n G_n = 0$$

Replacing $a_2$ with $a_2 +d$, this can then be iterated.

Hope this makes life easier ...