The pension was for his wife Sarah. He died before the first pension act for the War of 1812 for service alone so he couldn't have filed for one. You can get the pension file and the Bounty Land info from NARA.
This is what is on the pension application index card: Luke Bradley Bounty Land- 147-320-12 Widow- Sarah Bradley W.O. 33644 W.C. 34623 (These are the pension application numbers)
Service- Pvt Capt John Jones Co. 39 US Inf. Enlisted- Jan 23, 1815 Discharged- Apr. 30, 1815 Remarks- Dischg. certf. in brief. Residence of widow- 1878 Barry Co. ( P.O. Washburn) MO Maiden name of widow- Soldier's first wife, Agnus Woodall. Sarah Bryant Marriage of soldier and widow- Dec 8, 1824, Sumner Co. TN Death of soldier- Oct 29, 1862 Barry Co. MO Death of widow- about 1891