Is There a DIY Way of Fitting This Attachment to My Bathroom Tap?
You have obviously tried to no avail at hardware stores and the like. Somewhere adapters are available but sometimes it's exhausting to locate them. There are rubberized couplings for small diameter pipe similar to rubber drain couplings. 

Here's an example of a 3/4 inch one. 



https://www.homedepot.com/p/3-4-in-x-3-4-in-PVC-Flexible-Coupling-P1056-075/100187989