I would like to be able to do this in general, but I do need two specific dates.
What I need is the date July 17, 187 BC according to the Roman Calendar used at that time. I also need to know what date they would have considered to be exactly one year prior that on both their calendar and ours.
I hope that makes sense. Here are the two questions reframed:
On July 17, 187 BC, what would a Roman have said the date was?
What would one year prior to that date, according to that calendar, have been?
Julian and Gregorian dates cannot be converted to the republican Roman calendar, nor vice versa.
The republican calendar used mostly similar month names to the ones still in use today, but the length of the year fluctuated unpredictably. There are very rare exceptions, where specific dates are linked to dateable astronomical events: but they are very rare. For example, Livy mentions three solar eclipses: Livy 22.1.8 puts an eclipse to sometime shortly after 15 March 217 BCE by republican reckoning, but it fell on 11 February by the retrojected Julian calendar; Livy 37.4.4 has an eclipse on 11 Quintilis/July 190 BCE republican = 14 March Julian; and Livy 44.37.38 has an eclipse on 3 September 168 BCE republican = 21 June Julian. There are a few other isolated examples.
You could I suppose take the second of Livy's eclipses and use that as a best approximation for converting between the two calendars. But, and this is a very big but, the year was not 365 days long but 355 days; and even more importantly, the length of the year was unpredictable.
The republican calendar sometimes had an intercalary month of irregular length inserted into the last week of February. The Roman pontifices determined on an ad hoc basis whether there was going to be an intercalary month, and how long it was going to be. And evidence relating to the period 150-46 BCE is clear that the pontifices routinely manipulated this intercalation for political ends. They presumably did so earlier as well. See Samuel, Greek and Roman chronology (1972) pp. 162-164 for references and discussion.
As a result there is no knowable relationship between the beginning of the republican calendar year and the solar year. Sorry, you're going to have to wing it!