Let’s change the problem a bit: assume you’re starting with nonzero capital c, so the formula becomes (bt+c)e^(-at). If c>b/a, the derivative of that formula at t=0 is negative, so you need to stop immediately. That shows the decision to stop doesn’t depend only on a and b, but also on current capital. So basically “at each moment in time you face the exact same problem” is wrong. The naive solution is the right one: you should stop when c=b/a, which means t=1/a in the original problem.
Let’s change the problem a bit: assume you’re starting with nonzero capital c, so the formula becomes (bt+c)e^(-at). If c>b/a, the derivative of that formula at t=0 is negative, so you need to stop immediately. That shows the decision to stop doesn’t depend only on a and b, but also on current capital. So basically “at each moment in time you face the exact same problem” is wrong. The naive solution is the right one: you should stop when c=b/a, which means t=1/a in the original problem.