Let's try to put it down in less chaotic way.
As two observers observe the same sequence of physical events the only thing they can change is their representation of this sequence i.e. coordinate system.
Relative time dilation between two observers is real as we can establish delay in sequence of signals with static distance. So first of all we have different scale for time dimension for two observers.
Distances should be the same as speed of light locally is changing in a way that is consistent with unchanging distances.
Result of this is that orbital speed around MW for observer on Mercury is faster by the same factor as time is delayed.
Now we calculate GM-product μ using the same formula for both observers.
As speed is faster for Mercury observer but distances are the same for both observers we have that GM-product is bigger by that speed scaling factor squared. As GM-product have dimensions of time squared in denominator it seems that we have consistent picture so far.
Now if we assume that G is the same for both observers then mass unit for Mercury observer should be smaller by speed scaling factor squared.
This seems plausible as lowering mass in gravitational potential should convert part of the rest mass into kinetic energy.
Does this reasoning seems fine?