We were in the PNW in the summer of 2022. We rented a place near Rainier (closest "major" town is Yelm). I suspect we were farther away from Sea-Tac than you are. It was a rural, quiet location, but logistics were difficult and time-consuming due to the distances involved.
My husband and I each had to take early flights from Sea-Tac to Tucson a couple of weeks apart in June. In both cases, we spent the night before our flights at a hotel on International Blvd. (99) nearby with shuttle service to the airport. In both cases he had a delicious meal at 13 Coins (highly recommend it) right across International Blvd. from the airport. The 1st one was with me before he dropped me off and the 2nd one was with our friend from Bonney Lake who drove all the way down to the house to pick him up and drive him to the motel.
This was not the 1st time I or we've spent the night at a hotel near the airport before a flight the next day, even when we lived much closer or the flight wasn't early.
Since flights to Europe are typically later in the day, one thing you might check into is driving to Tacoma or Auburn and taking the Sounder from there. Both have substantial parking facilities. I have no idea what the cost is or what the security is like these days. Decades ago, we did park at the one in Tacoma over a long weekend one time when we took Amtrak to Portland and back. Others may be able to suggest other good Sounder parking options closer to Olympia.
It would probably also be worthwhile to check into any deals that the hotels might have that would include parking while you're gone. We did that a couple of times near Sea-Tac, but it was also decades ago.
I do understand the reticence about paying for long-term parking. When we're up there I'm shocked at the daily cost, especially since I pay $3.59 per day for covered parking with shuttle service to and back from the Tucson airport. I have to park there for 5 weeks before it costs about the same as a round trip shuttle from my house and back.