In general, it's fair to say that a company that doesn't have a UK office will not have a UK geographic number. However, this company appears to be different. There is one entry in the database which gives 020 7099 3787. This number is answered by an IVR (menu) voiced by an Irish accented lady, the same as is on 0844 330 2581.
On the Argus Care Hire website, selecting either the Ireland flag or viewing the
contact page gives an Irish geographic number, (023) 888 3002. To dial this from the UK the number is 00 353 23 888 3002 and you may find a provider such as 1899.com useful. This number is also answered by the same IVR as the UK 020 and 0844 numbers.
Googling the UK 020 number shows it is given on a number of websites, presumably operated by the same company:
The 0844 number is also given on the BookOnGlobe website (
www.bookonglobe.com). This gives the strong suggestion that they are linked.
The 023 number in Ireland is the area code for Bandon, County Cork which is about 10 miles from the south coast and no where near Dublin. However, Car Rental Town and WhatCarHire.ie give the number (01) 4999 600 for Ireland, which is a Dublin number. I haven't tried this so I can't comment on it.
I have updated the database entry with the UK 0844 number and the Ireland 023 number.