23 seats? That government could easily fall into a minority should one of their own become Speaker. Would I be correct in assuming Labour has the support of Progress?
Oh certainly! But Progress does indeed support them. Not in any official capacity, but will help carry most bills, if need be.
What is the ideological difference between the Liberal Conservatives and the Progressive Nationals?
The Liberal Conservatives are a merger of the former Liberal Party of Rhode Island and the Conservative Party of Rhode Island. The two merged with the intention of bringing a "moderate" force to Rhode Island to fend off from Labour, which was viewed (rightly) as a socialist party. The LCP's party platform is often centred around "big tent" issues that can unite everyone in the province, finding ways to reduce spending without hurting the working poor, working on 'sensible' regulations, and stressing a good working relationship with both Connecticut and Plymouth, with whom the province has the most economic partnerships with. The Progressive National Party is much more right-wing (comparable to say, a Midwestern U.S. Republican) and are wholly dominated by the "business" aspect of things. They are a cadet branch of the National Party, which at one time held seats in the Parliament but has since been supplanted by the Conservatives and Parti Francophone. Despite the "Progressive" in their name, their list candidates are quite right-wing and sometimes hold strong opinions on social issues (eg gay marriage, lgbtq+ rights, abortion) but the business-wing of the party makes sure that's never in the official platform.
The LCP and the PNP have effectively governed Rhode Island as a centre-right (by New England standards) government, providing little change but strong stability. Rhode Islanders are inherently hostile to leftist ideas, as Gina Raimondo is certainly well to the right of the national Labour party.