Chosen Solution
Since the battery replacement requires taking out the motherboard, does the motherboard need more adhesive to reattach or does it just stick to the old adhesive ? Just to clarify, I mean adhesive for the motherboard and not for the front panel, But does the iPad Air 2 adhesive strips come with motherboard adhesive? Like these btw iPad Air 2 Wi-Fi Adhesive Strips
The adhesive under the logic board is usually good when the board is re-installed, unless you really mangled it up a lot during removal. As for the screen adhesive, the one linked isn’t really that good. You should buy the Tesa Tape , also sold here. For roughly the same price, you’ll get more adhesive which can be used for other projects and the Tesa Tape is pretty much the best you can buy currently. You may also want to check out a previous answer of mine regarding some iPad re-assembly best practices (After doing my repairs (battery and screen) I have problems).
You should use new adhesive because especially on iPhone home buttons, when the adhesive meets a heat gun, it loosens and looses it’s use. You should definitely replace the adhesive.
So in short term, seeing you replaced the battery, I’d say if you used around 3 strips evenly spread along the battery it should be fine and stay held in, if you used less or prefer adding more to make sure it stays there feel free :)