Yes and no. For any vehicle that is carrying humans you'll want sufficient fuel to compensate for error or unexpected complications. And depending on the situation for a lunar landing that may require sufficient fuel to abort a landing well into an attempt so that a second landing (or return) can be attempted (instead of committing and hoping the crew just doesn't die). This doubly so if there's a capacity to refuel in orbit for a second attempt.

Also you'll need to assume for at least the foreseeable future that any Starship HLS landing will require sufficient fuel to return to orbit.

With enough margin left you can very reasonably adjust the center of balance by using multiple tanks and pumping all the fuel into rear tanks.

That should be enough given that the Starship HLS is estimated to have around 1.5 million kg of fuel at max capacity and only 100k kg of payload to the lunar surface (200k kg payload max for non-landing starship). That makes the payload mass to lunar surface only 6-7% of the total fuel capacity.

So outside of an extremely risky "attempt landing with no fuel left for an abort or return to orbit", you'll have at least double to quadruple the weight of the payload in just fuel alone.

Now for an earth landing of course the calculus here is different, especially since Starship's earth landing strategy explicitly requires throwing the vessel on it's side during the "bellyflop" and only pulling out of that fall with a powered landing at the last possible second.

TLDR No. Where center of mass will be able to make a difference fuel will make up significantly more mass than the payload.