Hi Jim, welcome to the boards. Couple of things to note (this is all UK specific, the US is slightly different)
1) Fun leagues rarely pay out cash prizes. Most of them will give out trophies (usually including such things as Individual High Game, High Series, High Average as well as Team Game and Series, plus top 3 placings) - If you want to recoup some of your expenses, you'll have to find slightly more serious leagues where cash prizes are paid. Downside of course is if you pay out cash at the end of season, you attract the better bowlers.
2) Cost of a league is invariably around the £10 per person per week mark (at least) - I bowl a scratch league which is £11, which is low (a lot of them are more than that). The centre takes about 85% of that as lineage for the games, with the rest going as prize money (or trophy money), plus expenses to the secretary and treasurer (who get paid a nominal amount for collating the scores each week, often 10-20p per bowler per week).
AMF aren't the cheapest of places to bowl either (as you've noted) - It may be there is a cheaper centre nearby (can't remember if that's the case with Washington - go to BTBA centre search page
3) Some centres offer cheap practise rates for league bowlers. It's worth asking if that's the case where you are. Most of them require presentation of a BTBA sanction card, so make sure the league you bowl in is sanctioned to get one (cost is £15 per year I think, payable in September)
4) The US league format is much different to the UK. They tend to bowl summer and winter seasons, whereas we bowl all year round (with splits for Xmas/New Year, and sometimes for August holiday). The US leagues also tend to "split" the league season into halves, with the winner of each half bowling for the championship. In the UK, league length is usually twice or 3 times the no of teams (for example 10 teams would be a 30 week season, 16 would be 32 weeks). There are exceptions, but that's a good rule of thumb, and it's like the football season, ie, winner of most games wins the league.
I may have missed some stuff, but that's an idea. It's not as expensive as you suggest, but you won't see much change from £350 a year I'd guess. You're right Golf can be cheaper (I play that too), but at least bowling can be played all year round, irrespective of the weather (ever had to abandon play due to lightening ? I know I have
For what it's worth last season, playing a scratch league, cost me £5 for a 48 week league due to the prize payouts. I know I couldn't have done that playing golf, as I'm only an 18 Handicapper, but the 215 average in bowling made it worthwhile
Hope this helped.