Hi James,
You could try a very thin bluish outline on the white text only, or then make a copy of the text as a sharp bluish “dropshadow”. In the latter case I would make it down and to the left so that it looks like the letter has some (but not a lot of) 3D thickness rather than a dropshadow (You may have to do it for both words for it to look right).
You’re correct to observe that the dropshadow and the reflection do not go well together, as the reflection gives the impression that the letters are standing upright on a reflective horizontal surface, whereas the dropshadow makes it look like they’re laying on a flat vertical surface, so with the dropshadow you are collapsing the depth you are gaining with the reflection.
You should fix the kerning on “York” while you’re at it. “Y” and “o” are too far apart while the rest are too tight.
And, getting picky here, the text should be higher up in relation to the center point of the frame so it doesn’t look like it is falling off the bottom. Visual center is higher than geometric center.