I vote yes on lavender. I suggest you conduct a trial by purchasing three or six lavender plants in small pots - 3 to 4" of a few varieties and plant them in an area that you can enjoy but also will not be devastated to see some fail. I also am of the belief that small plants establish more readily than larger ones and soon catch up to their larger, more expensive brethren.
Those that do particularly well can then be propagated via cuttings from the mother plants and planted out the following year.
Good drainage is more crucial than much of the other soil consideration. Here we have very heavy clay soil but because there is quite a lot of limestone the heavy soil still drains and lavender copes with it surprisingly well. Some varieties even self-seed for me. As for the comment about mowing vs. weeding, once established lavender can likely outcompete many weeds. You may want to intersperse with other sun loving drought tolerant plants as well and might consider exploring some types of thyme for ground cover and weed suppression. Many of the 'Mediterranean' herbs would grow happily together.
