Your lawn, your $, but I thought I'd let u know that the clover you're trying to kill is fertilizing your lawn for free. Clover is a legume. All legumes fix atmospheric nitrogen directly into the soil. The broad leaves of clover shade the soil surface, reducing soil temperatures, which help the soil retain water.
I've intentionally planted clover seed along with my grass, and I never have to use synthetics, or water as much as my neighbors.
Clover has been vilified by the fertilizer companies. I welcome it with open arms.
I am on the same page as you. Bought my house about 10 years ago. Worst clay soil I had ever seen. Hard like concrete all summer and swampy all winter.
First thing I did was stop using synthetic chemicals. Over the years switched to organic fertilizers, added compost, peat moss, etc. Slowly got better but still wasn't great. Two springs ago I planted white clover in the worst problem areas and it established very well and stayed green all summer. That fall and the next spring I planted more in other areas. It's a game changer.
This spring my lawn came in great without any applications and has continued to get better all year. Water retention is way better and doesn't get swampy. The clover is choking out the weeds and grass is growing again in the area i planted clover first. I haven't watered the grass all year and haven't applied any fertilizer in two years. The soil is getting darker, loamier (is that a word?), and the clay is disappearing. It's amazing. The wildlife on my property dwarfs my neighbors. The animals know the difference.
You have to be patient and live with weeds for a while but I will never buy fertilizer again. Best part is knowing my family is going to be much healthier because of these decisions.