So you are the other one I've been seeing drive around town. LOL. Just kidding. I'm from Madison.
I drive a 2007 Mercury Mariner Hybrid and replacing the battery was one of the first repairs I did (I believe the very first was the HV battery blend door actuator).
I replaced mine with a Optima Yellow Top Battery. I'm pretty sure lots of people will agree that it is overkill since it doesn't use the battery to start it the engine (it just to power the electronics in order to start the vehicle). My justification is that depending on driving habits, the FEH will use the battery in more of a deep cycle fashion rather than a battery used for starting current. Deep cycling a high current battery will kill it quick. An Optima Yellow Top battery is a AGM battery with both the cranking power of a starting battery with the deep cycle tollerance of a deep cycle battery. And it is maintenance free.
But, it is roughly twice the amount of the OEM battery. With the driving characteristics you described, I am surprised that it lasted as long as it did. The other reason I went with the Optima Yellow Top battery is because once the battery builds sulfate deposits, its capacity weakens and it frequently draws power from the inverter to charge itself, putting stress on it. This stresses the internal plates, causing them to warp and short together which drops the voltage. The charging system of the vehicle tries to charge the battery to its proper charge voltage but the battery has lost a cell and therefore it can only charge to 10 volts rather than 12. Once the battery reaches its max voltage, the rest of the energy is converted to heat. This will completely destroy the inverter in the FEH (to my understanding, it isn't cheep).
Optima Yellow Top batteries have a different design to keep this from happening. The do not have plates suspended in acid. They are sprial wrapped, like standard AA, C, or D batteries, and the cells are completely isolated from each other which keeps this from happening (protecting the inverter).
The decision between the batteries is completely up to you. I would not jump it and try to get it going because the battery is still dead and this will cause lots of stress on your inverter (although it is cold out so it should help keep the inverter cool). Buy the battery and a charger and charge it up all the way. I use a Battery Tender, 1.25 amp 3-stage charger which can be purchased from Batteries Plus and sometimes at departments stores like Walmart. They are under $50. Also buy a wire brush to clean up the terminals, the connectors on the car, not on the battery (they make sprays to help clean it to which is a good idea because it keeps the dust from becoming airborn while you are cleaning).
With the battery charged, unscrew the battery terminals from the old battery (remove the negative terminal first, then the positive), remove the old battery, clean the terminal connectors, insert the new battery (paying close attention to the positive and negative connectors on the battery (match red with red and black with black), connect the positive connector, double check that you just connected the positive connector to the positive side of the battery, connect the negative connector and you are set.
Final note, your vehicle will have to relearn the idle characteristics. There is a section in the manual that explains the steps you need to take in order to make this happen. Skipping this will cause your vehicle to idle oddly.
The above is the way I replaced mine. You could also just buy the battery, remove the old one, put in the new one and you are set. No charging or cleaning, but that is your call. Still, pay close attention to the positive and negative terminals. A handyman can change it.
Yes, they are heavy and can be dangerious to take on a bus (if they allow you to). The stock battery has liquid acid in it. The Optima Yellow Top is a non-spillable AGM battery and is safer, but still heavy and packs a lot of power.
Hope this helps.