I don't have a template or example of a business case, but I do often use a few simple tools to organise information to support decision-making.

Cartesian Coordinates is a model I was taught for doing change work with individuals in my NLP Practitioner training, to test the impact of change. I especially use it when coaching people who need to make a decision and they are struggling to see what the impact will be. I just call it The Decision Tool. It's very effective because it looks at an issue from every angle and it facilitates a focused brainstorming (great with groups). See attached for a template.

SWOT analysis (Strengths, Weaknesses, Opportunities, Strengths) can sometimes provide useful supporting information, too.

There are numerous articles out there about ROI for safety, which is relevant for making a decision to add resources. ROI Institute has some free resources, including some case studies and a book (albeit a bit expensive!) on ROI for safety. https://roiinstitute.net/case-studies/

Complete Learning (Beryl Oldham) offers some examples of case studies: https://www.completelearning.co.nz/case-studies/

In the past, I had the reverse situation, where a manager wanted to shed one person from the Quality testing lab. The technicians initially freaked out, but I told them we would just record everything we did for a month and total it all up. Then I gave it to the manager and explained that this was what we were doing with the resources we had, informing him that if he took away one of my technicians, he would need to decide which of those functions would no longer be carried out. His idea was very quickly dropped!

In your case, you might need to do a little work to identify the opportunity costs and the risks of what is NOT being done because you don't have enough resource to do it, and set out details of the activities the person would be responsible for carrying out.

In my experience, managers are on their own track with business activities; unless we go to them with sufficient thinking, preparation and data-gathering so we get their attention and they can clearly see everything, they are more likely to decline the request rather than delve more deeply into it. Try to anticipate from their perspective what it will take to make the idea crystal clear, compelling and undeniable.

Good luck with your mission!

One more thought.....consider starting with a meaningful, well-formed higher level goal that an additional safety resource will allow you to achieve, and build the case from there so that the additional resource is part of the means to the goal rather than the aim. That approach might get more buy-in.