I answered this question myself.
While I have found no precise way to calculate batch mean standard error, I have found that you can get precise measurements of perplexity in "mdl.FitInfo.History.Perplexity"
Therefore to calculate batch mean standard error:
% Input Initial Burn
burn = 100;
% Calculate Batch Size (b), number of batches (a), and true burn (whatever is left from a*b)
Matrix = mdl.FitInfo.History.Perplexity;
[m,n] = size(Matrix);
b = floor(sqrt(m-burn));
a = floor((m-burn)/b);
burn = m - a*b;
% Burn initial values
Matrix(1:burn,:)=[];
% Create Groups (G)
G = [];
for t=1:a
temp=t*ones(b,1);
G=[G;temp];
end
% Calculate Batch Means
Y = splitapply(@mean,Matrix,G);
% Calculate Batch Mean Standard Error (stdev)
Ybar = mean(Y) * ones(a,1)
Err = Y-Ybar;
VAR = b/(a-1) * sum(Err.^2);
stdev = sqrt(VAR);
