It wouldn't just be the Republic where it would cause problems. When I used to take my daughter to school in Belfast, at 0840 on winter mornings it was just getting light. Remaining on BST would mean it wouldn't get light until 0940.Most people who whinge about daylight saving time and campaign for BST all year round live not only in the far south of the British Isles, but also the far east. This means it stays light far later in winter than it does in say Derry or Lerwick. Before I retired, I was very glad not to have to scrape ice of my windscreen when I got up in the morning and set off for work at about 0830 as we were on GMT (in Cheshire).
Changing the system in the UK could also cause problems in Ireland where there would be two different times zones unless the Republic reluctantly followed suit.
Advocates of continuous BST just show themselves to be insular Southerners.