You don't do a monthly flat rate billing just to get a better income, or lock the customer into a fixed rate even if we have a drought.
You should only bid flat rates if you are in the business for the long term. But in some areas, flat rates are the norm, and in others you could never get a client to pay a flat rate.
I will offer a client a flat rate after three years of service. After 3 years, I can tell you within 15-30 minutes how much time I will spend on total service on your property. And since I know what I need hourly to charge for for the various services, and the average time for each of those services, I can easily fix a flat rate for a property. Just be sure to list what services are included.
Now the reason you need to be in the trade long term: you may charge for 30 mowings, and do 25, or 35, because of unusual weather. You don't refund to customer, or charge for extra mows, because you sold a flat rate service. (Hopefully you are providing a quality, dependable service, and that is what the customer is buying, not a certain number of visits.) Since you are there year after year, and you track the numbers and times, you can adjust your flat rates if actual number or times of service are changing in the long term.
Now the first year I included snow removal in the flat rate for some clients, I got blasted. Did 50% more snow work that year on each of those properties than the average for the previous 5 years. But since I'm in this for a long time, I was getting paid for that winter over the next 5 years, because flat snow rates are calculated based on previous 5 years' average. Have even had a few winters where I was paid for twice as much snow work as I actually did (I hate these, means I have to make up and work extra in future years, LOL).
Put whatever you want, and whatever the client wants, in the flat rate contract: mowing edging, trimming, shrub work, mulch renewal, spring-fall-general cleanups, fert & weed control, weed control in shrub beds, snow removal, anything. Just track your times, and make sure you are charging enough. Of course, if you're charging too much, someday soon you will likely lose this account.