Not the TNG bridge, but...
That is perfect. Looks as if right out of the work called "
The Doctor and the Enterprise."
Smith on the JJ Prise would be a nice fit too--
Here's one for you--the Narada wreckage lands on pre-tech Galifrey, explaining why Romulan consoles and Tardis work stations are similar...
Have the Tardis materialize in the shuttlebay,or better than that, have the chameleon circuit repaired, and as the bridge turbolift doors open, they step into the TARDIS there instead.
Enterprise needs an instant escape, you see Scotty attach a conduit to the light atop the Tardis, and the bridge and lower sensor dome on the bottom of the saucer flashes on and off as the Enterprise disappears from warp with the Tardis whirring sound.