I will preface my reply with sharing that I do not have any direct experience with your issue so if someone else has other input it would be helpful.
Google is trying to help users by offering the most common spellings of words. It is based on popularity in the search engine you are using.
If you use Google.com, the results will be different then using google.ie. For example, go to google.co.uk and type in the search term "cent". You will find the first suggested result is "centre" which is a british way of spelling "center". In Google.com, "centre" is not going to ever be suggested. You would need to type in exactly "centre" to see those results.
Bottom line, you want to spell your site name and terms using the language of your target audience. If you intend to split your focus between Ireland and the US, you should focus both spelling variations. If you have the resources, you can make separate pages for Ireland and US the same way you would for English and Spanish.