The Infinity's are a good choice. I use Alpine at work and while they are a good speaker, I really enjoy my ride home listening to my Infinity's. What you will need to do is pull out one of your rear 5.25"s and see if there is enough room for 6x9s. I cant remember, to be honest with you, if there is enough room or not. If so, yes the 6x9"s will sound much better because of the bigger size.
The thing about Bose, is they are enclosure people, not neccessarily speaker people. Rip apart any Bose home enclosure, car audio enclosure, Wave Radio, etc. and you will find out how well built the enclosures are. They spend their time and money building outstanding enclosures that make a small speaker sound big. The speakers really arent very good for the most part. Its all in the enclosure they are setting in. Since they usually consist of one speaker the range of music isnt really there.
There is an old saying about them: "No highs, no lows, must be Bose". They are great in the midbass, midrange and upper midrange, but the extreme high and low frequencies arent there. This is due to the enclosure giving a solid midbass, but is limited on the low end. Do this, put the settings in your moms car to the center (bass and treble), listen to a track you are familiar with, then go in and listen to the same track in your house system. Particulary listen to the extreme high and low frequencies.
In your case, the convertible is going to be the hardest obstacle to overcome. Since you have a dedicated sub system you should be able to get that volume ok. The hardest to overcome really is the midbass. That is hard to get in a car anyways, and being an open-top is way worse. The bigger the midrange speakers the better, and a midrange amplifier would be a great benefit as well. The panasonic speakers arent really the best for midbass anyways. They do good for the upper midrange and the tweeters are actually a little loud for my ears but they do ok there too. The Panasonic headunits really dont possess the power you need to overcome your convertible-ness
Doing a set of Infinitys (or Alpines, whichever sound the best to YOUR ears, not neccessarily mine, your buddies, the car audio saleman's...you get the point) and a small midrange amp would be my advice to you.