All you need to do is gets the appropriate tee for the job.
This tee allows you to connect two compression output to one input
![enter image description here](https://i.stack.imgur.com/pfujX.jpg)
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Turn off the water and open the taps.
Unscrew the copper and gently bend it to allow space for the tee.
Attach the tee and connect the copper to the tee at the most convenient branch.
Attach the braided stainless to the other branch.
Open the valves, check for leaks, close the taps, check for leaks and done!
You specify that the location is a laundry room.
I wonder if there are hammer arrestors (newfangled) or dead air spaces (oldfangled, may need draining the pipes once in a while to maintain) on the waterlines serving the washing machine, a prime culprit of high flow rates combined with quick-acting solenoid valves for the finest production of water hammer any household appliance can manage (equalled but not exceeded by the dishwasher, in some cases.)
Water hammer (whether or not it rattles the pipes so you hear it) is a sudden spike in water pressure caused by the inertia of flowing water in the pipe, when the place it's flowing to is quickly shut off. As a somewhat flexible item in close proximity to the end-point of the washer lines, the faucet hoses might be getting stressed by the repetitive pressure pulses.
A water hammer arrestor is basically a very small pressure tank, located at the end of the line to provide a compressible (gas/air) buffer. Water is effectively incompressible, and as such the pressure reached in the absence of an air chanber or arrestor can be quite high.
While it may be comforting to think that the high end faucet maker means only the highest-quality components, it may not be true. In this case, you may simply be exceeding the rated pressure on a brief, but regular basis - until failure results. If, perchance, the hose was kinked or in any way damaged (possibly in shipping and handling rather than though anything you did to it) that would make a weak spot in the hose.
Best Answer
If there are no usable threads on the end of your faucet spout, then, short of rigging something up with a piece of rubber hose and some hose clamps, you can't use that style of handheld bidet.
Instead, perhaps consider this type: https://rinseworks.com/product/abt-360-the-aquaus-360-handheld-bidet-for-toilet-nsf-certified-3-year-warranty
It connects to your toilet's water supply, rather than your faucet. Not only is this a more doable solution in your situation, but it will allow you to continue normal use of your faucet.