The OED mentions an example of "blow" meaning "fellate" from 1933.
The joke however is slightly older. It appears in the 4/25/1926 issue of the Oakland Tribune (p. 81), where it was credited to the Rutgers Chanticleer, a college humor magazine (see this source on Chanticleer). That version reads:
"A wealthy young lady named Fleau / Had a poor but good-looking beau. / Said Fleu to her beau: / 'Will you geau to a sheau?' / Said the beau, 'If you'll bleau I'll geou, Fleau.'
Since college-age kids would be more likely to use new slang, it is possible that "bleau" here has a sexual connotation. But notice also that this more original version has a rich/poor theme absent in the Consolation version. Since the beau was poor, and yet boyfriends were expected to pay for their girlfriends, the joke would make a lot of sense if "blow" meant "pay". And according to the OED, this was indeed an older slang sense of "blow", i.e. "to lay out or get through (money) in a lavish manner, to squander". Here are some examples it lists:
1874 HOTTEN Slang Dict., Blew, or blow,..to lose or spend money. 1892 Daily News 5 Sept. 6/3 Sometimes you'll blow a little money..but another week you may make a lot. 1896 Dialect Notes I. 412 ‘To blow oneself’, to spend money freely. 1904 W. H. SMITH Promoters v. 100 The Church people in England were the folks that had the money to blow. 1932 H. SIMPSON Boomerang x. 244 A thousand pounds, which she proposed..to blow in a couple of months' high living. 1957 Economist 21 Dec. 1030/1 He will probably feel able to blow with a clear conscience the £2,000.
Today, we may still say, "blow your wad" to refer to the same thing, but I think the intransitive use is obsolete, which is what occurs in the joke. That is why that this sense is not really accessible anymore, and the sexual connotation is the one that comes to mind more readily. But the rich/poor theme in the original joke clearly points to a sense of "spend lavishly", which is probably what was intended in the republishing of the joke in the Consolation (although the rich/poor theme was no longer overt in this version). This leaves the question of whether there was a double entendre in the original joke. If "blow" did have the sense of fellate at the time, then this is certainly possible. It is somewhat unclear however whether the old guys at Bethel (C. J. Woodworth at the time was 68 years old, I think) were up on the latest slang.