I think the electrosome H-bridge should be taken as a 'Ladybird books' level introduction to the circuit. When you've only got 6V to play with, you cant afford the Vce_sat drop of bipolars, let alone the Vbe drop of high side NPNs.
Take a look at TI's DRV8838 (or 8837) low voltage MOSFET H-bridges. They should be a good match to your application. Unless you are set up for SMD reflow assembly, you'll need a breakout board. e.g.
https://www.pololu.com/product/2990A PIC makes sense - just make sure you use a modern XLP one if its using batteries. The code will be nothing special - you could just let it idle along from its low power low speed oscillator at about 7.5K instructions/second, with a 1 second sandwich delay superloop and count loop passes for all timing etc. The low power INTRC oscillator typically isn't very accurate but I doubt you'll care miuch if there's a +/- 20% timing variation. If you need better accuracy, use the lowest divider from INTOSC, but then you'll probably need to implement some sort of power saving.