1 development team, multiple products, 1 product owner / backlog per product?

Personally, I think your suggestion is possible but I don't know if I would recommend it.  Your statements about there being two different arms of the organization with their own products makes me feel that there would be a lot of pressure put on the development team.  When you have multiple products it is difficult to say that only one of them will get any attention for a period of time, especially when one of them is in the growth stage. 

This statement from the current revision of the Scrum Guide seems pertinent

The Scrum Team is responsible for all product-related activities from stakeholder collaboration, verification, maintenance, operation, experimentation, research and development, and anything else that might be required. They are structured and empowered by the organization to manage their own work. Working in Sprints at a sustainable pace improves the Scrum Team’s focus and consistency.

Since a Scrum Team is responsible for all of the activities mentioned a single team of developers would be dealing with twice the work.  Even when they are allocating their sprints to Product A, they would be needed for activities on Product B.  For example, what happens if there is a bug discovered by Product B's users that is preventing them from being able to do certain work while the development team if focused on Product A?  

I believe that you need two teams of developers, at a minimum, in order to effectively create and maintain two different products. One Scrum Master might be able to support 2 teams but Product Owner and Developers should be focused on a single product. 

if we are working on product 1 for 4 sprints, product owner 1 would be part of the team. If we are then working on product 2, product owner 2 would be part of the team.

How long would it take before each product is worked on again? For empirical process control to be established and maintained, it should be possible to effect each release of a product at intervals of no longer than one calendar month. Any longer than that, and the feedback loop is likely to prove too attenuated.

Thanks for the advice.

Unfortunately we will not be getting any more resources. The development team is limited to 4 staff members. 

We are in a world where the same 4 will be working on multiple products and priority is up business needs with input from the team as to what is critical or not.

I prefer the idea of 1 PO for the team where she can handle all the competing requests and prioritise accordingly.

The issue I have is that she sits in our Operations area, the development team is in IT and the other stakeholders are in Policy. The organisation is very particular about their divisions. Ideally if the PO could sit with the rest of the core Scrum team that may make it easier for her to manage all the demands.

The organisation is very particular about their divisions. Ideally if the PO could sit with the rest of the core Scrum team that may make it easier for her to manage all the demands.

That sounds reasonable. How particular is the organization about wanting change to happen?