Breeding a bigger cucamelon

We grow a few cucamelons every year, sometimes from tubers but ours died in a harsh winter in spring so we reseeded, and with great suprise they had some small amount of genetic diversity, and 2 of our 4 plants had truly gigantic cucies. Like 2-2.5 inches which is almost double what most images i see of others, and what our regular ones where. I took seed and am awaiting 2024.
That got me googling and seemingly cucamelons arent yet cultivated/bred, i couldnt even find wild varieties of the plant. So ill give it a go and perhaps create the first, at least in english, cultivar of a giant cucamelon. Maybe I can get even longer ones with F2.
If i succeed, what should i call it? Maximelon? Cucamighty? Anyone know to what rate they self vs cross pollinate?

 

Rather than going for size, I think I'd start with getting more flavour and making the growth habit less rampant.

 

Once was quite enough, although I might have another go for the benefit of the chickens. They love cherry tomatoes.....

 

Yes but no named cultivars or variants.
Unfortunately they all had a similar taste and growth habit: I may be able to select some from this years genes but im also growing from tubers to see if they produce earlier. I do like them as a snack, but then we eat a cucumber a day normally.

My fantasy was a perennial cucumber, theyre more like a wierd goey green watermelon though.

Strangley enough this is the only crop that has ever volunteered in my garden. It produces enough fruit so it makes sense, if I had a bigger gene pool (and garden) I could breed for a british climate.