Sponsorship through either BA or AerLingus are worth exploring.
If you do decide to carry out some flying at a club, you might as well obtain your class one medical without delay.
The worse aspect of an integrated course for you may involve having to unlearn any horrible habits/techniques you pick up at a flying club. Choose your instructor carefully. The initial 10-15 hours are critical in building a solid foundation, and learning to trim, and TRIM PROPERLY.
No flying experience is necessary, and many say better, for the integrated course.