battery will be a lot more expensive and you are buying into a battery system
Screwfix/B&Q own make, lots o fcheaper makes, BUT i have no experiance of the cheap makes, I'm pretty much all dewalt for my DIY - as i wanted to last for some time,
sometimes the shops Aldi/Lidl do them - I would suggest saxtonblades for blades etc
The likes of screwfix and toolstation sell corded oscillating saws for about £40 (300 watts).
I guess they are fine for light work. On the balance of probability, they will be noisier and vibrate more than a pro version.
Both companies offer (or did) full refunds in year one. If it fails, you take it back, ask for a full refund and purchase another... (thereby giving you another year of warranty).
I don't know what a mini version is though. I have never seen one.
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