I'd vote for the Pioneer system with a non-Pioneer sub. I've listened to the Andrew Jones BS22's (the bookshelves) quite a lot and A/B tested them against two other mid-fi, extremely nice bookshelves (the Ascend Acoustics CBM-170 SE and the Axiom Audio m3v2). The Pioneers, while very (very) slightly lacking in bass compared to these two more expensive speakers (each about $300), are on the whole comparable. They have a sonic quality very much like the Ascends, which is one I really love for music. For $125 they are incredible speakers. The BIC DV62si has, like Jones' speaker, won a lot of praise over the years as a great bang-for-buck bookshelf. I have never heard them, so I won't weigh in - but I've read a lot about them over the years, and when the DV62si has been actually measured by third parties, as I recall, it does well for the money, but it does not measure up to the Jones' speaker for accuracy.
The Jones tower is almost a clone of the bookshelf, with more bass range and the height advantage of the tower construction. I have not heard the center, but I have to assume it has a strong sonic resemblance to the BS22's. In that case, it's got to be an excellenct little HT center. The frequency range of the human voice happens to be the sweet spot for Jones' gem of a speaker.
You can pick up a substitute sub without adding much or any cost to your system. I'd go with one of the Dayton subs at Parts Express, assuming you don't want to raise the total cost. Unsolicited advice: If you think you want a better sub ($400 or up) down the road, try to get it asap without spending $100 now on a second-rate sub. That will be $100 down the drain.
Final nods for the Pioneer system: looks, music, and flexibility. The Pioneer is no cherry wood beauty, but it's not bad looking at all, and better looking than the BICs, unless you go for the big box BICs (and have a fond nostalgia for 1970s big boxes). The Pioneers will likely give you better response on music, in stereo or 5.1 mode, and that's a huge plus. Finally, with the Pioneers, you're getting speakers you will likely always be able to find a place for in a secondary system (bedroom, study) should you catch upgraditis and end up wanting to drop $3K-plus on a new system some years down the road.
I've often been curious about the BICs. My family had BIC America 3-ways in the 1970s, and I kept them until they blew out in 1990. I always loved the warm sound, deep bass, and the little dial I could use to adjust the ferro-cooled tweeters. But I'm pretty sure that today you'll be getting a higher quality product with the Pioneers. They are not $800 speakers, but they're uniquely enjoyable for their price range, lacking the mistakes (extended highs, boomy bass, bad crossovers, lack of accuracy) of many comparably priced speakers.