- Mixer capability - The Evolve M series seems to offer the most inputs. I have a full featured 20 input digital mixer that I can always use.
It's best to make that decision up front: to mix externally or internally. I generally choose to use the most powerful mix solution - which in your case would be the Allen & Heath rather than the speaker. If your doubling their role for use at home - then maybe an internal mixer makes sense. In a rental circumstance the internal mixers are a source of error and abuse.
- Sound quality - I don't know where I can compare these.
It's not really necessary unless you are trying to combine them with some other brand. Your audience will not be doing a comparison test, and how they ultimately sound will be equally as much a function of the space they are used in. If you like the JBL for example, you'll find a way to make it work for you.
- Battery power - If I go this route, my choices are limited and I must sacrifice output power.
Professionally, we have to have a really good reason for using batteries. Wireless mics are the conspicuous example. The need is continuous, conspicuous, and without parallel. If you are not already faced with consistent regular work in areas with no other reasonable power option - then you get the battery model. Otherwise, what's the point?
Convenience is an illusion. The battery product will be heavier, but smaller, and lower powered. Old batteries are inconvenient or costly to discard, and replacement batteries are often no longer supported in favor of the next generation. (Changing battery form factors is how manufacturers drive new sales. If they change the shape or voltage of the battery - you need a whole new speaker!)
- Bluetooth - Should support v5.0 or higher and hopefully aptX-HD or LL codecs. Even better if it supports LE Audio but I doubt any do yet.
Internal Bluetooth presumes you'll have discreet access to your speakers should you need to reset, recover, or lockout your link. I typically use external receivers that feed into a system via the XLRs so I can locate the receiver in a more convenient and discreet location.
If I had speakers hidden at the front of a presentation upon which I would not want to intrude - then BT receivers are actually located externally where I could conveniently resolve any issue without distraction. This is also helpful in any installation where height would make built in BT reset or lockout controls inaccessible.
Truth is, I never use Bluetooth AT ALL for any program specific content. It would only be deployed to allow a client side content add, or background music. I would never use Bluetooth as a path for example, during a wedding procession.