The dress is fine for Basic Skills and probably is nicer that what some of the skaters in Tot events might show up in. The key to Basic Skills competitions is to have fun without a lot of pressure.
I'm going to go against the majority here and say, no, she doesn't need private lessons just to compete in a Tot or lower-level Basic Skills competition as long as it's the elements event. If she wants to skate a program, to music, then yes, she will need some one-on-one time to learn the program. My club encourages our group lesson skaters to enter our Basic Skills competition every spring and the coaches will generally organise a little mini-group lesson duirng the post-lesson free time for the skaters who will be competing, just to run through the elements and the way they will have to perform them at the competition. If she wants to skate a program to music, then a weekly 15 minute lesson for as many weeks as it will take her to learn the program and some extra practise time would be good - generally it's three to four weeks prior to the competition but that depends on the skater since some take longer to memorize programs than others.
As for club membership, if she's enrolled in a USFS Basic Skills program, she will have a Basic Skills membership already, either through the rink or the club that hosts the lesson program. Many skaters I know ( and mine is a low-level club so I know a lot of them) stick with group lessons until they get through most of the Basic badges and then slowly transition to private lessons once they are in the Free Skate Badge area. She won't need a full club membership until she gets into testing Pre-Preliminay Moves In the Field which is a ways down the road at this point.
I'm not going to get into the which rink program is better issue. All I'll say is that bigger isn't always better.