Have you not got any girls?
I haven't done Beavers for years, I do Rainbows these days, but my sister does Cubs, so hopefully i can be some help.
Yes you can join with other groups for trips, sisters cubs often join with the same groups beavers and/or scouts so that could be an option too.
Trips wise, it doesn't always have to be expensive, if you have a school with a young farmers group they'll sometimes do quite hand on sessions cheaply (my local one charges £1.50 per person) fire station do free tours, tesco do a free thing called farm to fork, they take them on a tour behind the scenes usually do some food tasting, give them a goody bag and a badge. Metro bank do an evening where they'll show you behind the scenes, again free and with goody bags at the end. hikes, nature reserves and trips to the park are usually a hit too.
Not cheap ideas - tobogganing has been my Rainbows favourite activity so far, you will need 1:1 ratio though as Beavers are too young to go on on their own
Acceptable price depends on your area and how affluent it is, plus what you are doing, we generally subsidise trips from unit funds and try to keep it under £10, unless it's something really special.
Bring a friend night sounds good, you could also try asking local schools if they can let you do an assembly (maybe wait til you've been there a bit first) and advertise that you have spaces in school/church/community newsletters and any shops that will stick a poster up for you.
Are they doing a badge every week? or did i misread that? everyone loves a badge apart from the parents sewing them on but you need to remember that they are meant to be a challenge so maybe stretch them out a bit, and that badge costs start to add up. also that there are only so many badges and you don't want to end up having to repeat things.
glad to see Scouts conning people in with 'it's only an hour a week, i thought it was just us Guiders getting caught out on that one 