People always flame me for saying this, but if you aren't in the best of cycling shape already, then riding more often might strengthen your legs up pretty quick and a lower gear ratio will be useless to you.
Also, I think you'll be going outside the specs for your current rear DR and while everyone agrees Shimano specs are conservative, you might not exceed those without some other stuff besides just a new cassette and chain.
More sprockets on the front will give you the most range of gearing over anything you can put on the rear. However that also will be an expensive upgrade.
I'd probably just opt for a smaller ring on the front for the near future and see what that does. Depending on the crank, that might be the cheaper option and more effective at getting you up hills comfortably. Your high end will suffer some, but probably not as much as you think since it is unlikely you always ride in the 11 cog and at your maximum speed!
Maybe you need another bike.
Welcome to BF.
Last edited by Iride01; 07-04-22 at 11:45 AM.