While the external monitor itself is not the cause of the 'weakened' battery, leaving your MacBook Pro on AC power all of the time will definitely affect the battery's performance. It is essentially under a constant state of charging, even though this is regulated by the charging circuit, which will eventually cause the battery's chemistry to begin to break down.
Current batteries will typically last the longest when used regularly, but not completely discharged with each use.
Check the ac/dc adaptor with a multi meter it should match the writing on the unit. they can run hot and burn out it might be possible to have 2 broken units.
disabling or enabling devices that use power can be used to diagnose a battery problem e.g. blue tooth fire-wire thunderbolt SD CD and DVD drives
Disabling OR enabling as many as possible and changing the settings to low power or high performance should make it last longer or less long. either of these situations confirm a battery problem in a case of less or more charge. a logic board problem would make no difference if you get my drift. The idea is a normal battery would not suffer much impact like this in short term use but a dodgy battery would be very sensitive. Its not really relevant if you can't boot because of no charge.
I've had battery problems before these are my steps I know you have probably tried already but if your desperate it might be worth trying again.
take the back off your mac and check to see all the ribbons are undamaged also check to make sure the the ribbon connectors are not cracked twisted or broken its a common problem when unplugging the battery to damage the connector they are very delicate and require a very light touch.. When unplugging them you cant just flip them out they need even pressure from each side and the middle and gentle levering very tiny amounts at a time if you are in the habit of flipping them out you may have inadvertently damaged them, if you think you may have been careless with the connectors you can get replacements inexpensively and this is a common issue on the mac.
i would disconnect the battery from the logic board and leave it for 15 mins then be very careful to firmly plug it back in using even pressure so all of it is plugged in evenly, be very careful. don't connect the mag. (trying to discharge the mac battery while the mains is plugged in won't discharge the battery). Then close your mac back up and screw it back together. before you plug it in the mag hold down the power button to 15 seconds or longer, something should happen a beep, a flashing light so just wait. if after a while nothing happens plug in the mag. But just wait for something to happen, mac batteries have an issue where they can get all screwed up and need discharging and this is how I would fix it.
the main reason for a battery needing this is usually due to hardware changes (for example recently I'm betting bet you have made some hardware changes like an SSD or just better memory), the mac will work fine for a while after a hardware change then need a negative discharge, its a one time deal and due to a new power setup from voltage changes in the hardware. Replacing the battery won't change the situation as the system still needs to reconfigure its new voltage settings. Don't try to reset the battery with the mains pugged in its all done without any external power, i think its a lot to do with the boot up process and probably unique to MBP.
---edit i edited this to make it more informative and corrected spellings---
Best Answer
Charging the MacBook Pro with the lower wattage from the display is not bad for your battery health at all.
There's also nothing wrong with the computer consuming more energy than the charger delivers when under heavy use. It won't damage the battery nor the charger. Ofcourse it will eventually discharge the battery so that you need to lower the energy consumption in order to actually charge the battery.
Note that a MacBook Pro does not usually use near 97 watts all the time when in use. It is usually in short bursts if at all.