In all honesty, given your budget constraints your best bet is to stick with your D3100 and kit lens, plus supplement with one or two good bang for your buck lenses from the second hand market. Unless of course you can sell your D3100 and kit lens, then some options may open up but I doubt you'd get much for it unfortunately.
In good light, your D3100 and kit lens will still perform well. I don't know the exact kit lens you have but most are decent optically but may be of poorer build and not particularly fast.
You could consider adding either a 50mm f1.8G or 35mm f1.8DX depending on the FOV you require. I think the 35mm f1.8DX will be a more versatile focal length and open up your shooting opportunities in lower light situations however the longer effective focal length from the 50mm f1.8 gets you closer to the more classic portrait focal lengths.
In regards to the 10-20mm, no compact camera will give you that sort of FOV so you're really comparing very different capabilities even if they are around the same price. 10-20mm on APS-C is in the ultra-wide region.
So you need to decide on what capabilities you want.
Lastly, just a thought but with regards to video you could consider just using your phone camera for this, depending on what phone you have of course.
EDIT: In addition, you could consider going over-budget for the trip to get the equipment you want but sell off equipment you don't need after the trip. But to make this cost efficient, buy used and stick with just lenses. You don't loose as much on a lens (if at all) if you buy and sell used.
And one more thought. If you are just a casual photographer, I think you might be better off with just a better zoom than your kit lens. Eg. a Tamron 17-50mm f2.8. Again, consider buying used and offloading after the trip if you don't think you'll use it much again after the occasion.
