Find constant that allows an integral to be finite

Another approach, taking for granted convergence at infinity, is to find values of c that annihilate all terms of the series expansion of the integrand at r == 0 of the form a*r^-n:

Series[(Exp[-r] (1 + f[r, c]))^2*4 π r^2, {r, 0, -1}]
SolveAlways[Normal@% == 0, r]

Probably, extended comment.

This integral is of the form:

Integrate[Exp[-2 r] r^n,{n, 0, Infinity}] =
ConditionalExpression[2^(-1 - n) Gamma[1 + n], Re[n] > -1].

Now let's look at our integral:

Collect[ExpandAll[(Exp[-r] (1 + f[r, c]))^2*4 π r^2], r]
Cases[%, num_ r^n_/;n<0 :> num]

We can solve this for c:

Solve[# == 0, c] & /@ % // Flatten // Union

{c -> -3/2, c -> -1}

We can check these and make sure that only c = -3/2 leads to convergent integral of π/16.