Firstly, the use of the word 'dick' to mean 'penis' is attested in the Oxford English Dictionary in what appears to be an erotic novel published in play form in 1890, and the OED also testifies that the term is used in Henry Miller's novel Tropic Of Cancer from 1934 in a context referring to circumcision. There appears to be some suggestion that it is a fair bit older than 1890, too - sexual slang terms are often a fair bit older than their first recorded published use, due to taboos and prohibitions and so forth.

Additionally, the phrase 'Tom Dick and Harry' goes back to the 18th century at the least; the OED records a speech by a J. Checkley in 1730 that uses the specific phrase, and it appears to be the case that other groups of seemingly generic names have been popular at other times (e.g., 'Tom, Dick and Francis' may be an earlier version of the phrase).

So yes, 'dick' was very likely in common use at the time of Cole Porter's 1948 musical Kiss Me Kate (which features the song 'Tom, Dick and Harry'), which you can hear a version of in the 1953 Hollywood musical version here. In the musical, the song is sung as part of a sort of play-within-a-play - the character Lois Lane is singing the song as part of a musical production of Shakespeare's The Taming Of The Shrew where Lane is playing Bianca.

In context, it's probably about as risque as I'd expect from Cole Porter? As a lyricist, he's definitely a member of the Tin Pan Alley club that celebrates wordplay and clever rhymes. And they were generally quite happy to sneak in knowing glances at sex they can’t quite say aloud (e.g., Rodgers and Hart have lyrics in Pal Joey about an affair). By 1948 it had been a couple of decades since Porter had written ‘Let’s Do It (Let’s Fall In Love)’ where, let’s face it, nobody is meant to believe that ‘it’ really means ‘falling in love’.

In terms of the rhymes in the song in your quote, you can see him moving around the order of 'Tom Dick and Harry' in order to rhyme 'qualm' with 'Tom', and so, thus, rhyming 'double quick' with 'dick' makes sense. In the context of a musical version of Shakespeare, I think some mildly racy wordplay might be justifiable, too - ol’ Willy S. wrote lines about ‘country matters’ in Hamlet where from context it’s clear that the matters in question are focused on the first syllable.

And I mean, sure: if you remove some words with an ellipsis, the single-entendre is clear: 'I'm a maid mad to marry/ And will take double-quick / Any....dick'. But there's quite a lot of words in that ellipsis nonetheless; it feels like the kind of reference that would be played very straight on a kids show, but with a knowing wink to the parents in the room rather than a single entendre.

So it's more double-entendre than obvious single-entendre, but I think the entendre would have still...entendred. Ann Miller playing Lois/Bianca in the Hollywood movie clip I've linked above plays it relatively straight - though the flash of upper thigh right after the first ‘dick’ might not be a coincidence. I suspect more than a few people playing the role have put just the right emphasis on 'dick'.