We recommend that guests fly into Toulouse (TLS). From there, Siradan is a ~1.5 hour drive. Direct flights to Toulouse from the US might be scarce, but one-stop connections are plentiful through other cities such as Montreal, Amsterdam, Lisbon, Madrid, Brussels, Frankfurt, Munich, London, Geneva, and others. Due to the Olympics, we would recommend against connecting through Paris.
If you're already in Europe, several budget airlines offer flights into Tarbes-Lourdes-Pyrenees Airport (LDE), which is even closer than Toulouse, but will require some logistics work on your part. Other potential cities to fly into include Pau (PUF, ~1.5 hour drive), Biarritz (BIQ, ~2.5 hour drive), and Bordeaux (BOD, ~3.5 hour drive).
An option for the adventurous is to fly into Barcelona (BCN) and drive approximately 5 hours to Siradan. If you choose this option, there are a few things to consider. First, the drive is primarily via a two-lane mountain road, which means the views are gorgeous, but the driving itself can be a little frustrating - we've done this trip and would only recommend to experienced drivers. Second, we would recommend that you spend some time in Barcelona and plan on taking 2 days for the drive, stopping to stay at Parador de Lleida on the way. Lastly, you should plan on returning via Barcelona as well, since returning a rental car in a different country in Europe typically comes with expensive one-way fees ($1,000+).
Flight prices can vary widely depending on the airline and class of service. More established carriers are advisable over budget airlines and Google Flights can be a valuable tool for comparing prices and flight options.