Nick is a former colleague of mine and his history of the 9th RIF is superb. As you can see the NIH were drafted into the battalion alongside soldiers from many other regiments so I think you can say with near certainty that they were rebadged in the same way that all of the other transferees would have been.
https://www.nickmetcalfe.co.uk/the-e...th-irish-hors/
Several photos of former NIH soldiers wearing RIF badges here:
http://www.northirishhorse.com.au/NI...age/People.htm
And the 9th NIH in 1918 -every man wearing RIF badges.