You will pay UK PAYE and NIC on your salary and BIK under UK tax rules in the UK.
As an Irish resident you will also be liable to Irish tax on this income, but may be able to avail of cross border relief if you qualify for that.
The BIK on the company car will be calculated under Irish tax rules if taxable in Ireland, which will be different from the UK tax rules.
I'm not sure about Customs issues and VRT - ring the VRT office for this information.
Whether or not to take the income or the car will have to be worked out, based on the figures involved and your tax position, so I cant comment on that.
As you have trans-border issues I would suggest that you get professional advice.