I don't recall any incompatibilities between the two gate networks being mentioned in the series. And we do see direct galaxy-to-galaxy gate travel on multiple occasions throughout SG-1 and Atlantis, prior to the construction of Midway, which would seem to rule out this explanation entirely.
However, do recall that Midway is, well, the midway point on a series of gates that "bridge" the vast intergalactic distance between Pegasus and Milky Way galaxies. While not technically necessary (as we've seen Jack gate to the Asgard's home galaxy, and of course the Atlantis expedition got to Pegasus just fine without the bridge, among other examples), the power requirements of intergalactic gate travel are enormous -- remember how big a deal it was to get the Atlantis team there in the first place, and that for quite a while they had no way to get back, nor could Earth do it again to send supplies or even talk.
The gate bridge was built to solve that problem. Designed and programmed by Dr. McKay (probably with help, although you'll never hear him admit it!), it is most decidedly not the normal functioning of the gates -- each gate in series receives the matter stream, buffers it, and then sends it along again. This allows travel along the gate bridge with only the power requirements of normal interplanetary gate travel!
Midway serves two purposes stated in the series: Security (a little C4 in the right place, and the bridge is better than an iris!), and decontamination (why they never worried about this until they built the bridge though is beyond me...). We do see the security functionality in play when the station is taken over by the Wraith.
And now we're finally to the dual gates in Midway. Remember that the whole bridge is a non-standard re-jiggering (that's the technical word!) of the normal gates. Basically they've been reprogrammed by McKay et al to function outside of their normal parameters. So the gates could do whatever he wanted them to! It seems that the gates at each end of the station were "hard-wired" to only be able to dial the next gate in the bridge. Ditto each gate in the bridge -- depending on direction of travel, each gate is hard-wired to dial the next in sequence. Thus the closeness of the two Midway gates is irrelevant -- McKay just programmed them that way. It wouldn't be the first time SGC personnel have tinkered with the programming of the gates, although for once it didn't cause more problems than it was meant to solve!