Welcome to the forum.
I have a history (as crew on the larger craft) of boating up to 50+ft over many years and now I'm quite content to get my time "at sea" with a sub 4m SIB. Great fun at minimal cost.
You are smack bang in the most popular size/price requirements so may have to look hard everywhere and jump on anything that looks suitable.
Do read post #1 in the Which SIB thread... Which SIB, which SIB, which SIB!
Here's an example at £1550 start and no bids yet. Alloy floor which can be a faff to assemble but more rigid feeling than an air floor. A very very sought after outboard with it...
Inflatable Boat | eBay
Size wise a modern 3.2 like a Honwave can feel quite safe but older small tube/low bow models not quite so good. 3.5m is a good size to aim for... 4.0m can get a bit heavy unless trailered.
This time of year with the sun out your ideal outboards are reaching the start of peak season money. A very clean last of the 15hp 2-strokes in lovely order will be £1200-£1300 so you need to get really lucky on a package to end up with a nice 15. You are right in my opinion to set 10hp as a minimum. But only go down to a 10 if you get a lightweight like the 26kg Tohatsu 9.8. Most of the other 10s are exactly the same weight as a 15 so go straight for the 15. 20/25hps oddly can be a bit cheaper as they are getting to a weight many don't want to carry about.
If you have any ability to tow consider buying a package with trailer as these are often the size you would want with a suitable 15-25hp outboard. You can then sell the trailer off and effectively lower the cost.
Hope that's given some ideas for you to raise questions.