newtonian gravity - Dimensional analysis - When can you introduce constants that make dimensions compatible?

I have just read this question: What justifies dimensional analysis. One statement was:

Maybe the speed of a comet is given by its period multiplied by its mass. Why not?

As a formula this is $v=mT$. How do we know that this is wrong? I am not asking for the standard answer concerning the incompatibility of the dimensions. Suppose $v\propto m$ and $v \propto T$, then one could argue that $v\propto mT$ and so $v=CmT$ where $C$ is a constant that fixes the dimensions.

I will give another example: $F_G=Gm_1 m_2/r^2$ - Newton's law of gravity. As far as I know, Newton knew the following: $F \propto m_1m_2/r^2$ and he didn't know the value of $G$ so he simply stated $F=Gm_1m_2/r^2$ with $G$ fixing the dimensions.

Now comes my real question: My intuition tells me that $v\propto mT$ is not right but can you exclude it with dimensional analysis? Then you would also have to deny Newton's law of gravity. How do you know that $v=CmT$ is incorrect but $F=Gm_1m_2/r^2$ is not? Especially: When can you "invent" a constant in dimensional analysis which fixes the dimensions? If you could do it all the time, dimensional analysis would not be helpful...