To save you the trouble of trying to find that specific battery, any standard 7Ah 12V SLA will work fine. I recommend AGM (absorbed glass mat) type batteries over Gel Cell batteries, but either will work.
It’s likely that nothing will happen. The thing you have to worry about is the fact that while you know the Ademco transformer is legitimate and has passed quality control measures, other off-brand transformers may be labelled as equivalent when in reality they fail to meet the voltage or current requirements.
For example, I have several Chinese manufactured transformers labelled as 24V. Two put out 24V as they should - both are brand-labelled transformers, one Telemecanique/Schneider Electric and the other RIB. The other two are off-brand - one puts out 34V under load, the other can’t quite top 21V even without load. Not to mention the two off-brands are not filtered.
When you stick with brand-name transformers, their quality control measures (and manufacturing practices…but I won’t get into that) keep you a lot safer.
You could try it. I don’t mean to be a stickler, but I still recommend a brand-name unit, if not the original unit. Try to see if you can pick up an Amseco XF-1840 or XR-1840LED transformer.
They’re two different companies, but with similar names and some similar products. Ademco is a Honeywell company, Amseco is now owned by Potter Electric Signal.
With the above in mind, most signals that satisfy the voltage and current ratings of the NAC will be fine. You’ll be safer using electronic horns, as mechanical horns tend to have high inrush currents and throw off back-EMF into the NAC circuit. Older panels were more apt to handling this, some newer panels don’t respond as well to these surges.