Seville to Lisbon

Fly. The airport in Seville and Lisbon are close to city centers, so it would not be difficult.

it's a tough call and one I've faced before. For me, the driver would be the best, by far, as you said, like a road trip, stopping for snacks, photos, etc. BUT the problem with that is that it will cost a fortune...if you are ok with the price, that is what I would do. To give an example, I hired a driver, arranged through my hotel, to drive me from Sevilla to Cadiz two years ago. That short (scenically boring) drive cost me 200 euro. Again, if you find a price you can accept, that is what I would do. BUT there is no guarantee that a driver will speak English and I think the odds are not in your favor. A guide, yes, but most drivers I've hired in Spain do not speak much English. For me it's ok cause I speak Spanish and have had some wonderful conversations with drivers in Spain, and in Italy, over the years.

Otherwise, just fly. What is it, Vueling? Iberia Express?

I've not looked into the trains and you don't say how long but it seems as if the length is putting you off. Flying, despite all the airport issues, at least is quick and usually painless. Why don't you check the trains again and let us know the length of the ride, and we can see where you will travel through.....might be good for scenery.

Each of those alternatives has merit, and downsides!

Not go from Sevilla to Lisbon, at least with limited time?
You can take a bus and break up the trip provided you have time.

If using public transportation, an option is bus from Seville to Faro (~2.5 hrs) and then a high speed train from Faro to Lisbon (~3 hrs). If you want, can just take the bus which would be ~ 7 hrs. I prefer trains, so if the connection works, I'd switch from bus to train. Could always add a few days to the Algarve if you have time to break up the trip.

I personally really like Seville and Lisbon, so thus why would fly and not lose any time getting between the two.

https://www.lonelyplanet.com/articles/seville-to-lisbon-by-train

As others have said, the train journey is awful - several changes and takes over 8 hours.
Flixbus has a direct service in about 6.5 hours, with four stops en route. There may be other buses that are shorter. \this is probably the cheapest option.
The car is shortest 4.5 hours, but the road are good - have you thought about hiring a car in Seville and driving, then returning it to Seville at the end?
Contrast the cost of that with 450-500 euros for the car plus driver.
And with the cost of flights. taxis etc.