I'm in the same boat...I have not settled on a Twitter app that I like. Twitter has taken a HUGE step back on Win10 Mobile. Official Twitter for Windows Phone 8.1 provided actual notifications from individual user tweets, just like on iOS and Android. Now for some reason, the "new & improved" Win10 official Twitter app lost that functionality, and I have to resort to ENABLING TEXT MESSAGE NOTIFICATIONS to get user tweets. That is ridiculous. Next step is for me to go back to the Win 8.1 Twitter app so that I get that function back. Anyway, enough of my rant...
I've tried the follow 3rd-party Twitter apps with limited success. All of them have given me notifications when I'm mentioned or get a direct message, but I can't comment on the custom sound for notifications:
Twittone
Aeries
MeTweets
Gleek
TweetIt!
I think all are free except for Aeries...
Maybe one of these will work for you MaryJane?