High end brdge, compete with DSLR for a lot more manual features; Low end bridge often have little or none. Whilst 'zoom' is over rated.
Brdge's offer a heck of a lot of it from the effective magnfcation of the crop factor using a micro-sensor, which also tends to increase DoF for any given focal length and make Auto-Focus less critical, hence 'cheaper'. Mega zoom's will also tend to have peculiarly restrictive max apertures at long zoom settings, that beg slower shutter speeds when you least want them.
On Crop-DSLR the kit 18-55 is my most used lens; on film, with a variety of cameras and lenses for them, I can say without a shaddow of a doubt that the 35mm mild wide, if it isn't the fixed focal length of the camera anyway, is where the 'zoom' setting so often ends up at or very close!
O/H's 21x zoom coolpix, has an enormous zoom range, and goes about as far or further than 300mm on wdgetal...and it is significant USELESS!
If that much zoom is used, light camera, and low shutters so often result in blurry shots, ether from camera shake or subject movement. And having that much zoom, on a button inclines folk to use it.
Moderate the amount of zoom you use, to get the wider apertures and faster shutters.... may as well not have that 'extreme'.. it's just wasted.
On a DSLR, you at least stand a better chance, with 'better' long zooms of not suffering so much shutter-speed drop out, from faster apertures, and you likely have a bit better chance of dialing in some 'help' from a higher ISO, and or better 'icon' auto or even semi or full manual exposure mode.PLUS heftier camera, and optical view-finder that can encourage better steadier hand holding, or even a tripod!
Atthat budget, if you have sold-on a D3300.. I would strongly recommend you go look at D3100 ad D3200 DSLR's
second hand; and take the hit of lack of zoom, for the advantages of a DSLR, and the lesser overall compromises elsewhere.
The D3100 even at a 'mere' 16Mpix is still more than good enough ad its 16Mpix are better mega-pixies coming from a bigger sensor than from a cheaper brdge's mico-sensor... and you have the versatility to get a longer lens later, or even upgrade the body later, rather than trying to get 'everything' in one go..... the Nik-Kit 55-200, is a nce partner to that camera, too, again, lackng the 'ultimate' zoom they are unlved, even new thay are a sub £100 lens and you should be able to pick one up for under £50
second hand... if you are lucky, you should come accross Dixon 'bundle' kits of D3100 18-55 and 55-200, for under the £150 mark, which wouldn't stretch your budget too much.
As to the battery issue?
For an electric everything, LCD screen only consumer compact, yup; AA's are likely advantage. NHm's of a decent Ah rating do last well, even compared to alkali's, but you still have the newsagent option of chucking in disposeables at a pinch.
I have four Lith-packs for the DSLR... and a 12v cigarette lighter charger for them to plug into car or bike when out and about! I don't want to miss photo-ops for lack of battery.... and O/H and daughter having D3100's and nicking MY lith-packs can make that a TAD more challenging... but still
Would't be big issue in the purchase decision which sort of battery either took.. other than with a bridge, I would have to factor a set of NiHm and a charger into the cost on top of ask-price; for a DSLR, I'd expect to get a mains charger and at least one Lith-Pack, but would likely be budgeting for a couple of spared and a travel-charger off e-bay anyhow.
A-N-D, I wouldn't be in big rush to get as much zoom as I could.. I really wouldn't.... I have rarely used much more than 2000mm, and for twenty years that was on film cameras where cop factor shrinks that to about 140mm on a crop sensor DSLR! And having 'useful' reach is far better than having unhelpful 'maximum' reach.. and seriously unless you are birdig and need to fill the frame with a sparrow at a range of five miles, you really just DONT need go that long!
