Lufthansa checked bags.

So long as your Seattle-Frankfurt-Milan is booked as a single ticket, your checked luggage should meet you in Milan.

Make sure any medicines and other things you might need during or immediately after the flight are not in the checked bag. Just in case something doesn't make it there the same time you do.

IF you have to gate check your full size (and overweight) carry on at FRA, at least your bag first made it across the Atlantic and is in Europe. And if it's gate checked, at least your bag made it to the correct gate and aircraft. These two factors minimizes the chance of a mishandle. That said, it would be wise to carry immediate essentials, valuables, fragiles if any and documents in you personal size bag if you can.

And BTW, if possible your deensest (max weight/min vol) in your personal bag or jacket pockets to get your carryon as close to 8 kg as you can. And don't walk around the gate with you backpack carry-on looking like an overloaded donkey. If it looks light it's less likely to attract attention.

Lastly, Lufthansa usually allows one checked bag at no extra charge. You can always check a bag with your heavy less essential stuff and not sweat the go or no go.

I have checked my bag all the time for years and years with no issues. That being said, make sure you have in your personal item all electronics and chargers, keys, financial information, and medicines just in case. Anything you might find irreplaceable and maybe a partial change of clothes. Be prepared for the worst, but expect the best...as that is the likely scenario. Again, I've never had an issue.

There's been at least one story on this forum of packing car keys in his checked bag on the way home and then the bag arriving late and the person being stuck at a hotel in the airport.

We also have checked bags for years and have always retrieved our bags straight through to destination. That said, on a straight through ticket, we look at the tag before it's placed on the conveyor belt. I've been known to question the agent to reiterate as well! (Peace of mind).

We travelled Lufthansa this past July. My counsin checked her bag but her carry on was a little overweight. They checked it for her no extra charge. Same scenario on the return flight.