Say there was a lifeform not unlike ours native to a planet somewhere in one of our nearest satellite galaxies, say Sagittarius, during their nighttime would they be able to see the Milky Way brightly? I can imagine it would be very beautiful.
It would be a border case. The Andromeda galaxy is on the border of the free eye visibility. It is enough visible that it was found by medieval free eye astronomers (but, for example, the ancient greek didn't see it).
Our galaxy is similar to the Andromeda, thus it is a similarly strong light source. @dmckee mentioned in comment, positioning yourself closer to a not point-like light source increases the raw amount of light arriving, but also increases the solid angle over which it is presented. Thus, our galaxy would be around similarly visible as the Andromeda for us, although it would be much bigger.
Note, being in a dwarf galaxy doesn't mean, that the star density in your local region would be small, or that the atmosphere of your planet is enough transparent for that.
The Apparent Magnitude of the Sagittarius dwarf galaxy is 4.5, and it is 10,000 light years in diameter. The diameter of the Milky way is 10 times that, so if we can see "them", they can definitely see us, assuming no dust cloud near their planetary system.
My thanks to dmckee for correcting this simplistic approach above to what apparent magnitude actually represents:
Your first paragraph is a non sequitur. You have to evaluate (in some approximate way) the amount of light coming from a patch of sky; that is the luminosity per unit solid angle as seen from the remote location and what you've written—neglecting as it does stellar counts and any other approximate luminosity measure—actually argues for a dimmer appearance.
If the putative inhabitants on a planet in the satellite galaxy are either extremely long lived, and/or their civilisation survives long enough, they should get a really good look at the Milky Way as time goes by, as their galaxy is on a collision course with the Milky Way,
Based on its current trajectory, the Sgr dSph main cluster is about to pass through the galactic disc of the Milky Way within the next hundred million years, while the extended loop-shaped ellipse is already extended around and through our local space and on through the Milky Way galactic disc, and in the process of slowly being absorbed into the larger galaxy, calculated at 10,000 times the mass of Sgr dSph.