will a stroller fit through the subway?

For a regular single child width stroller, there is nearly always a wheelchair barrier which I'm confident would fit a single width stroller.  A double width stroller may a tighter fit so I can't be certain.  But there is also often a fence barrier at one end of turnstiles / gates near ticket booths which is unlocked & opened manually by a station agent for allowing passage of extra wide mobility devices.  If this was to be used, you would validate your ticket at any normal turnstile (inserting ticket into slot, retrieving ticket from top or tapping a pass to purple reader) then returning to the gate opened by an agent to pass through.
 
Keep in mind this doesn't solve the problem of stairs, which there are almost always in the Paris Metro system.  Many of the older stations, of which St. Germain area would have, there are often small or large sets of steps you would have to navigate in between turnstiles and actual Metro train platforms.  Doing this with a stroller is not easy.  Often random strangers will offer you a hand to help carry the stroller up or down.  But, you may wish to finish the final portion of your travel to hotel after arriving at Gare de l'Est using Uber.