I would go for 9cm pots for the time being. I don't think the potting compost is too critical but if you need to buy new then something with a bit of JI (soil) or sharp sand mix.

yes I 'd also go for a 9/10cm pot but not much bigger. I usually mix 50:50 a JI type with compost to open it out a bit. lavender originates from the med area so good drainage is essential really.