If gravitational time dilation was in any way connected with the velocities of stars in the Milky Way, we should, in general, see a relatively smooth, monotonic curve. Gravitational time dilation scales as
where $r_s$ is the black hole's Schwarzschild radius and $r$ is the distance from its center. However, we don't see that. What we do see is a sharp increase in velocity from relatively low values near the center of galaxies to some peak, and then a gradual decline.
You've probably seen this occur on galactic rotation curves before; I decided to grab some data from this page. There's raw data for stars in the Milky Way here, given in kilometers per second relative to radius in kiloparsecs. You can see the peak appear in the data within about 20 kiloparsecs, and then a gradual decline.
I also plotted select data points:
Here they are with a polynomial trendline1:
Dark matter density profiles can explain the curve well; gravitational time dilation can't.
This is actually a very non-quantitative way of looking at it. I've just given a weak argument by stating that any effects from the supermassive black hole should vary differently with an increase in $r$. For a quantitative demonstration, I'd recommend looking at some of the answers in the linked question and links therein, including this one by Rob Jeffries and these two by John Rennie, which cover both types of time dilation.
1 A polynomial trendline like this isn't what would be used to model the velocity dispersion of stars in a galactic disk. However, I'm just using a polynomial fit (12th-order, because that's what's needed to reduce bumps) to better show the overall curve from the data points.