US car seats are not allowed to be used in Europe. It doesn't matter if it's for a single day as a tourist or for a longer time as a resident. No one will notice if you use your US seat though. Especially not in Spain where car seat habits are very poor.
Planning ahead for car seat safety during a trip is great and I wish more parents did this. Traffic is the number one killer of kids in most places. Using a car seat on the aircraft is irrelevant for safety and we have tons of data to back this up.
Even though it's technically not allowed to use a foreign car seats in Europe it can still be a great idea. Familiar installation and a car seat which a child is used to. Since you will be visiting many times it might be a better option in your case to purchase and leave a seat in Europe.
Since the very beginning I have placed car seats at places where we often vacation. We have seats in Mexico and also at our summer residence in Sweden. At both places we spend in total at least five months a year. First it was rear facing seats and nowadays high back boosters and low back boosters.
If you will be purchasing a European car seat the two best all-round seats in the world are Britax Hi-Way and Britax Max-Way. Both are light weight, compact, have the 55 lbs rear facing weight limit, have passed the ultra strict Swedish Plus Test to 55 lbs and also fit well in pretty much any car small to large.
HW will last to around age four or 110 cm, MW to age five or 115-120 cm. MW is kind of an upgrades HW with a bult in head support, integrated harness and a little taller seat sell. HW is the most compact rear facing seat with belt installation but MW only need a few centimeters more space.
When my kids were young we always brought two HW seats all over the world. Lots of trips to US, Mexico and all over Europe. We did 60+ flights in the first few years and the seats fit well in all kinds of vehicles. Limos, taxis, rentals, vans, friends cars, etc. Always checked separately in car seat bags with bubble wrap, diapers and extra clothing as protection. Never had a problem.
EU cars must all have Isofix from 2013 but it's of course possible you get a rental which is older. LATCH is compatible with Isofix. The LATCH solution is unfortunately not very good and too many compromises had to be made in the US version. The rigid Isofix with a support leg is easier to use.
There are a couple of seats, Kiss 2 and Triofix Recline, which can be installed with Isofix or seat belt. This means far faster installation but also the lower 40 lbs rear facing weight limit and higher prices. These seats will also fit in pretty much any car but the downside is the higher weight. These weigh 30+ lbs compared to less than 15 lbs for HW and MW. Not fun to carry around an Isofix seat.
Overall you will find that other mentioned above that car seat habits in Spain are very poor. Similar to US. Lost of unrestrained kids and basically no rear facing. Same general trend occur in Europe as in US. The further south one travels the worse habits get.
Just noticed that you will also need a seat in UK. Is it possible to take the train or public transportation? This will greatly simplify things and also keep it safe.
Don't worry about traveling alone with a child. It's no worry at all. I have done tons of trips alone, 20+ hours across the world, with two small kids. Select a night flight if possible, I also prefer the bulk head seats with more space, have plenty of change and diapers in the carry on and also of course something that keeps your baby calm. And bring some extra food along to go along with the breastfeeding (if applicable).
Enjoy your trip and the great Spanish food