Determine the Signal Curve from Parameters of a Power Curve by Noisy Measurement
Since your domain for the parameters is limited (Only 9 options) the best way for White Noise would be going through them and pick then one with the least Mean Squared Error (MSE) which is the parameter to minimize for AWGN.
in MATLAB it will be something like:
function [ paramAlpha, paramBeta ] = EstimateModelParameters( vT, vY )

vParamAlpha = [1, 2, 3];
vParamBeta  = [1, 2, 3];

bestMse = 1e50;

vX = zeros(size(vY, 1), 1, class(vY));

for ii = 1:length(vParamAlpha)
    currParamAlpha = vParamAlpha(ii);
    for jj = 1:length(vParamBeta)
        currParamBeta = vParamBeta(jj);
        vX(:) = currParamAlpha * (vT .^ currParamBeta);
        currMse = mean((vX - vY) .^ 2);
        if(currMse &lt; bestMse)
            bestMse     = currMse;
            paramAlpha  = currParamAlpha;
            paramBeta   = currParamBeta;
        end
    end
end


end

In a simple performance simulation I got the following:

Which means that for STD in reasonable value for this case (Since your maximum value is around ~3) your performance will be great.
The code is easily adaptable to any values of the parameters you'd like.
Though I think such brute force methods are reasonable for up to ~100 combinations.
Another thing you might try is use Affine Model on the log of the values.
Then apply some &quot;Rounding&quot; to the values. Though this will change the properties of the noise, in real world it seems to be effective way.
The full code is available on my StackExchange Signal Processing Q70753 GitHub Repository.