Green’s Dictionary of Slang

1. (Irish) a dog, usu. a mongrel.

1977		J. Plunkett Coll. Short Stories 81: ‘That’s the photograph you see in front of you,’ John Joe said. ‘The boys setting off to get the bowler’ .
1981		(con. c.1920) P. Crosbie Your Dinner’s Poured Out! 31: How I longed for a sheep-dog of my own, ‘Hey, mister, give us that oul’ bowler’.
2001		G. Coughlan Everyday Eng. and Sl. 🌐 Bowler (rhymes with Cow-ler) (n): dog/ugly person.

2. an ugly person [by ext. of sense 1].

2001

see sense 1.