You should not stay home! Don't miss a chance to go to Lake Como!
While traveling with a 2-year old overseas is a challenge it can be done - we travelled all the time when my son was little. We do have family in Germany and went nearly every year. But also went to France (18 months old), Spain (close to 4yrs), Scotland (6yrs), Austria (7) and Lake Como (9 years). All trips were great fun! You just need to adjust to the speed and interest of your grandson! Slow down and enjoy.
Lake Como is fabulous. A vacation spot for >2000 years. My husband and I went 20 years ago for a conference. for work. We went with our son in 2019, and only the pandemic prevented the return trip in 2020. From Milan you can take the train to Como, then take the ferry to your town / accommodation, especially if it's on the Western side. Easy connection from tran station to ferry by taxi (without luggage you can also walk). On the Eastern side, you can reach e.g. Varenna fairly easily by train, and take a ferry from there (we stayed in in Bellagio, which was reached easily; the apartment was fabulous and new on the rental market, and affordable. It costs a fortune now...)
Activities with a toddler: even my American ice cream rejecting kid LOVES gelato. You can go on a gelato exploration tour in the town you stay in. I am sure you will find many different gelaterias you can try. Visit a villa and garden, and enjoy the blooms of spring with your toddler. Combine with taking a boat tour. Try to identify all the fancy cars. Visit some churches and look at all the paintings and statues together as a "treasure hunt". go to a local market and admire the local produce and all the goods. You can find schedules online, I think; kids love the markets, and we don't have them like this here in the US (but with some exeptions). Even just having a very leisurely day on Lake Como without much sight seeing is a fantastic day!
On the hillsides above the towns on the Western side of the lake are also opportunities for hiking / nature. We had planned for that but it was a very hot July in 2019, and thus we stuck more to gelato (and didn't gain weight from all the walking up and down the hills)
Oh, and if you or your daughter are looking for actually well made toddler shoes that are not sneakers and designed for growing feet and reasonably priced - stock up! Italy is comfortable while fashionable shoe paradise - including for kids!
Overall, I'd agree, especially with a toddler, less is more! Take maybe one stop on route (by train) to Rome and explore for a few days. When the child is older, you can come back together and go to all the art museums then.
Happy travelling!