Your next door neighbours to the west have a parliamentary system of government with a popularly elected president.
There's no need to reinvent the wheel or for constitutional scholars to turn themselves inside out trying to understand how it could be done.
There is also the example of the Glorious Revolution, when William and Mary were in effect invited and then elected by Parliament to be constitutional monarchs.
In Ireland the popularly elected president has an appointed council of state to advise him or her and examines bills before signing them into law. The president (usually acting with the agreement of the council of state) and on behalf of the people can refer a bill to the Supreme Court if he or she feels there's a constitutional issue that needs examination. Money bills cannot be referred, nor can bills changing the constitution once passed by the Dail. The president can refuse to dissolve the Dail (parliament) when the Taoiseach (PM) has lost a majority/ vote of no confidence.
The president represents Ireland abroad for official visits and welcomes other heads of state. The essence of the role is that the president acts on behalf of the people vis a vis the government at home, and represents them internationally as a figurehead.
The current incumbent, Michael D. Higgins, was a political fixture for decades (Labour Party). He is a poet and booster of Irish literature and the arts and the Irish language, currently in the last year of his second and final seven year term. A presidential election will be held no later than October of this year.
People who put themselves forward for election as president usually have some party political background, but it's not necessarily always the case. To secure a nomination as candidate, an individual must be nominated by at least 20 members of Dail Eireann (the parliament) or four administrative counties or county boroughs. A sitting president can nominate him or herself to run again. Previous presidents include academic /constitutional lawyer from NI Mary McAleese, human rights campaigner Mary Robinson, and various pols.
Presidents tend to be exceptionally judicious in the exercise of their powers.