I have had lots of divided tanks, and with just one filter it has never worked for me. Every tank caught a bad case of velvet from the lack of filtration. I lost 5 bettas to it.
What I started doing was using sponge filters, and had one in each section. I have valve controls for each line, and just one air pump. It's worked beautifully.
I also want to advise you on the DIY dividers, which I have also done. Be very very sure that they are snug to the glass, even if you have to silicone them in place. I just had a very aggressive fish attack through the 1/16th inch gap between the glass and divider, and two boys are just torn to shreds. He also jumped to get to one of the fish, and my water level is 3 inches below the dividers. I have kept probably over 50 bettas and I've never had one so aggressive before.