@Alejandro Jakubi Thanks so much for you reply! Rouben Rostamian's comment was more along the lines of what I was looking for, but thanks anyway!

@Rouben Rostamian  Thank you so much for your awesome insight! I was just fiddling around with subs, but think I was a bit fatigued after having been looking at this problem for so long! This is definitely a trick I'll remember. Thanks again!

Sorry, but I'm not sure if the above commands are clear enough. The first, second, third and fourth commands from my above explanation are:

f(sqrt(m/(m+1))*(x+1/(2*sqrt(m))), sqrt(m/(m+1))*(y-1/(2*sqrt(m))));

mtaylor(f(sqrt(m/(m+1))*(x+1/(2*sqrt(m))), sqrt(m/(m+1))*(y-1/(2*sqrt(m)))), [x = x, y = y], 3);

mtaylor(f(sqrt(m/(m+1))*(x+1/(2*sqrt(m))), sqrt(m/(m+1))*(y-1/(2*sqrt(m)))), [sqrt(m/(m+1))*(x+1/(2*sqrt(m))) = x, sqrt(m/(m+1))*(y-1/(2*sqrt(m))) = y], 3);

f(sqrt(m/(m+1))*(x+1/(2*sqrt(m))), sqrt(m/(m+1))*(y-1/(2*sqrt(m))));