No, it’s fine.
Saying Tehillim arouses Divine mercy, and the mishebeirach tefillah afterwards directs Hashem’s rachamim towards the choleh mentioned. Hashem’s mercy is endless, so nothing is “stolen”.
In fact, we always include all cholim (“b’soch sh’or cholei Yisroel”). There’s nothing wrong with specifying your father’s name.
May R’ Belsky and your father both have a complete and speedy recovery, b’soch sh’or cholei Yisroel.