The global level counter resets after going from 0 through 4:
Language: asm
81B4: INC (IX+07h) AF:0044 BC:015E DE:03E8 IX:702E ; increment global level counter ($702E + 7 = $7035)
81B7: LD A, (IX+07h) AF:0000 BC:015E DE:03E8 IX:702E ; load the result to A
81BA: CP 05h AF:0500 BC:015E DE:03E8 IX:702E ; check if it equals 5 now
81BC: JR C, +04h AF:0542 BC:015E DE:03E8 IX:702E ; if true, jump to $81C2
81BE: SUB A, A AF:0542 BC:015E DE:03E8 IX:702E ; else, reset A to 0
81BF: LD (IX+07h), A AF:0042 BC:015E DE:03E8 IX:702E ; write that 0 to global level counter
When that happens you go back to level 1 ($7035 == 0), and barrels speed doesn't seem to change.
I player up to level 11 and nothing changes at all. Your speed is the same, enemies speed is the same, enemy spawn speed is the same. The game seems to track how many loops you've played, but I fail to find any difficulty changes.