I'm gonna offer a different definition from the 日本国語大辞典 that I think gets closer to the usage in this case:
Basically just a slightly old-fashioned/poetic way to praise something as beautiful or glorious, not necessarily at a peak time period (although obviously, pretty much by definition the two concepts are related).
Here is an example of 花の独身 from the 花の80s, allegedly an issue of Kurashi no techo from 1980 (the issue number does match up, but I found this in Google Books so don't cite it in your thesis without pulling the original...)
Quick and dirty translation:
"She is a 'glorious single' who changes jobs as she pleases, has traveled overseas multiple times and within Japan as she likes, just as it pleases her, dresses fashionably down, and roams the land eating well. I am a happy young wife living with a kind husband and two adorable children in a home we own."
I think this describes pretty well the kind of glorious, carefree life that 花の独身, if taken absolutely unironically, would imply (note however that the paragraph goes on to detail how the narrator spends all her time cooking and cleaning and dealing with neighborhood gossip, that her kids are exhausting, she has to wear ripped, dirty jeans, etc. -- so whatever is going on here, clearly it isn't just "how happy we both are!". Wouldn't be surprised if the next paragraph was about how the single woman has no savings, gets nagged by her parents all the time for grandkids, etc...)
If you search online you can find people asking "Do folks still say 花の独身?" and other people answering "No, because there are so many women who want to get married but can't nowadays."
So, yeah, I think that this person's bio says "hana no dokushin" rather than just "dokushin" because she wanted to say it in a self-aware and humorous way.