Step 1: The naming of the carpal bursae follows the side of the hand they serve.
Step 2: Radial corresponds to the thumb side. The flexor pollicis longus is the long flexor of the thumb, and its synovial sheath is precisely the radial bursa, which continues to the distal phalanx of the thumb.
Step 3: The ulnar bursa, on the medial side, wraps the eight tendons of FDS and FDP, so those two options describe the ulnar bursa rather than the radial one.
Step 4: Flexor carpi radialis lies in its own tunnel and is not involved. The thumb flexor therefore defines the radial bursa.
\[\boxed{\text{Flexor pollicis longus (FPL)}}\]