One of the products I have found over the years not to cause as much skin reactions as other detergents is Omo Sensitive Liquid.
Most pets will tolerate sitting on bed linen washed in this product; be that in their own pet bed or in their owner’s own bed. Currently the 2 litres ($20) on sale at Coles at half price $10 ($5/litre-which is cheaper than the otherwise normal best buy version which is the 4 litres for $24 or $6/litre).
Most sensitive skins do well with this cleaning product.
For a Rinse/Conditioner, normally we advise Cleaning Vinegar but as the human consumption vinegar is now cheaper than the cleaning vinegar-its fine to use that as the rinse/conditioner.
We don’t promote Cuddly or Febreeze as we have seen some very nasty reactions from both of those products.
Hope this helps reduce some of the allergen load that our allergic family members, 2 and 4-legged are having to deal with.