I had rubber backed area rugs on a laminate floor for years with no ill effects. Vinyl flooring, on the other hand, will stain from the rubber.

Ok...so engineered hardwood is an option because this is "ground level" and not below grade. If you do not want to heavily invest in the home's flooring (and after a big renovation you feel the pinch at every turn...I get that), then an excellent water resistant laminate (I hardly need to add "out of Europe"...) will give you an amazing floor for 15-20 years without much effort. Work with something out of Switzerland or Germany and you will be ASTOUNDED by the toughness and the handsomeness of the product. Quick....do it be Brexit! While all the tariffs are still "Zero"! I like water proof/water resistant laminates because you are in IRELAND...it rains there from time to time. Sometimes water comes in on your Wellies...and that just isn't good for regular laminate or wood. As water resistant as you can please. It will pay itself back 10 fold by the time you finish your second winter with your flooring.

Of course if the faux wood looking synthetic planks are VINYL (they fit your description as well) then the type of rug you describe is probably OK...Again...you have vinyl flooring planks, then you will need to look for the "Safe for Vinyl Floors" in the description.

I have laminate floors with large area rugs that add comfort and visual interest and I can take them to the cleaners. I would be concerned about a carpet remnant because the backing may not be meant to be on a finished floor. You might also not get the comfort since there is no pad underneath. Houzz and Wayfair all have awesome affordable rugs that are meant to be on top of finished flooring so less chance of discoloration of the floor beneath or of sticking.