Should you game while on battery mode?

It’s perfectly fine. Leaving the laptop plugged in all the time so the battery is always at 100% isn’t ideal, but it’s still better than using the battery when you don’t have to. If you want to maximize longevity, go into the BIOS and set the battery max charge to 80% and the minimum charge to 50%. That way when the battery reaches 80%, the laptop will essentially run itself on wall power but NOT keep the battery charged. Instead, the battery will be allowed to self-discharge over a period of many days down to 50%, and at that point the system will charge the battery back to 80%. This is best because the battery spends most of its time essentially doing nothing, neither being charged nor being kept topped up nor being actively drained. And limiting the charge to 80% is better than keeping batteries maxed out all the time, which is why Tesla cars default to an 80% charge and you have to specifically request a “Max Range” charge each time you want one. The downside to this strategy of course is that you only ever use 80% of the battery’s capacity, and if you set the minimum charge to 50% and need to unplug from the wall unexpectedly, you might only have a 50% full battery. Basically, you can optimize your battery for everyday use or long-term life, but not both.

Yes it does strain the battery. Discharging the battery at high current isn't long-term healthy. These batteries are prone to swelling after a year or so, and Dell often don't want to supply replacements.

If you want your battery to last long, also set max charge level to 80% or so when you don't need the battery. With Dell Command / Power Manager you can change to 100% in advance when you know you want the full charge.