Estimation of the amplitude of a frequency component in a signal
Generally, the frequency estimation problem decouples from the amplitude + phase estimation problem.

As I said in the comments, you could just do:

$$
\hat{A} = \left|\sum_{n=0}^{N-1} x_n e^{-j 2\pi\hat{f} n} \right|
$$
where $\hat{f}$ is your frequency estimate.

Regarding your issues:

because the number of points isn't high enough to hit the right resolution

I'm not sure what you mean by "resolution" here? Resolving the amplitude?

second is that I need a really fast way

The suggested technique will only take $N$ complex multiplications and $N-1$ complex additions, plus whatever you need to calculate the absolute value (two complex multiplications and one addition).

That's pretty fast, and linear in $N$.  And certainly faster than an FFT (which would be $N\log_2 N$).