DIY Computer UPS - Smoothing DPDT Relay 80ms Cutover
That is a bad choice of battery.  It's a starting battery, not a deep cycle battery.  You'll destroy it lickety split.  You may want to hang out with the off-grid homesteader / solar / wind crowd for awhile and learn how to set up systems like this.  

Also since you're willing to spend this kind of scratch on a battery, maybe evaluate some of those Tesla modules that are out there.  They respond much better to deep cycling than even a deep cycle battery, which this is not.  

80ms is typical of a mechanical transfer switch.  But further, you will have trouble because of it throwing automatically, and possibly having some false starts, as power cuts out and returns.  To be clear, 80ms is five full AC cycles - high low high low high low high low high low.  So there is just no chance of doing this with DC-style inductors or capacitors.  There's no such thing as a DC battery.  And even then, how would it be cut in?  The transfer switch's contacts are in transit. 

Really, you should be reinventing the whole kit-n-kaboodle to run on low voltage DC.  The server's power supply should input LVDC.  The router, switch, modem whatever should input low voltage DC (probably already are).  Essentially it will be hung off the battery at all times with makeup power coming from a battery charger.  

Now, you don't need a transfer switch.  

The kind of money you're spending on that gear, you could get a LV power supply for a server, and also a nice lithium Tesla style battery with some range.