Why should the last point be excluded when performing a least-squares fit of a periodic discrete time signal?
First (wrong) answer (for integrity) The $y$-value of the last point
is the same as the first one. As you apparently know the frequency,
this point comes in excess of the &quot;fundamental period&quot;. It sounds like
this additional point comes like an implicit double-weight to the
first  point of the period.

Second take: I have tried to fit the data, with or without the last point. It seems to fit well, both cases.

%https://dsp.stackexchange.com/questions/71164/why-should-the-last-point-be-excluded-when-performing-a-least-squares-fit-of-a-d
clear;close all
%% Settings
T = 1.5; w=2*pi / T;
t = (0:0.15:1.5)';
y = [2.200 1.595 1.031 0.722 0.786 1.200 1.805 2.369 2.678 2.614 2.200]';

%% Fitting all points
ft = fittype(@(a1,a2,a3,x) a1+a2*cos(w*x)+a3*sin(w*x),'coefficients',{'a1','a2','a3'},'independent', {'x'});
f = fit( t, y, ft );
% Plot fit
subplot(2,1,1)
plot( f, t, y )
axis([t(1) t(end) 0.5 3])
grid on
title('Whole point set')
%% Fitting all points but the last
ft = fittype(@(a1,a2,a3,x) a1+a2*cos(w*x)+a3*sin(w*x),'coefficients',{'a1','a2','a3'},'independent', {'x'});
f = fit( t(1:end-1), y(1:end-1), ft );
% Plot fit
subplot(2,1,2)
plot( f,  t(1:end-1), y(1:end-1) )
axis([t(1) t(end) 0.5 3])
grid on
title('Whole point set minus 1')