This is not an answer since I cannot give the references you are requesting for reasons that will soon be apparent but am not entitled to comment. Firstly, there is something fundamentally wrong with the framework of your question--there is no duality between spaces of bounded, continuous functions on euclidean space and locally bounded measures thereon. (Try integrating the sine function on the line with respect to Lebesgue measure). What one does have are natural dualities
a) between the bounded, continuous functions and the finite Radon measures;
b) between the space of continuous functions with compact support and the locally finite measures.
Note that it has long been known that in case a) one cannot use the norm topology for this duality since it leads to a much larger dual space. This was remedied in the 50´s by R.C. Buck who introduced a complete l.c. topology (the strict topology) on the bounded, continuous functions which has the bounded Radon measures as dual. He worked in the context of functions and measures on locally compact spaces but this was soon extended to that of general completely regular spaces by various authors.
In b) one uses the natural l.c. topology on the functions of compact support as a so-called strict $LF$-space (Dieudonne and Schwartz). This was the basis of the Bourbaki approach to measure spaces, defining them as the dual spaces of suitable locally convex spaces of continuous functions (a precursor to Schwartz´ treatment of distributions).
With regard to your question, let me start with the case of functions and measures on a compact subset. In this case there is a natural l.c. topology on the space of measures which is complete and such that the associated convergent sequences are the ones in your question. This is the finest topology which agrees with the weak star one on bounded sets. Despite this description, it is l.c. I imagine that it coincides with the one you describe but I haven´t sat down to check this. This applies to any compact metrisable space, by the way.
At the moments, due to covid, I am not in a position to give precise references but the relevant themes are the Banach-Dieudonne theorem and the bounded weak star topology. There is a reasonable article on the latter in Wikipedia but it is, rather curiously, in German (with no English version).
Memory suggests that they are discussed in Koethe´s monograph and Schaefer´s standard text.
In the non-compact case, the natural topology on the locally bounded measures would be a corresponding projective limit of such space. This is then isomorphic to a closed subspace of a product of spaces of the type described above which means that it will inherit many properties from those of the compact case. It is even a complemented subspace (in the l.c. sense) which makes the situation even more tractable. I haven´t thought about whether it can be described as in your query. I am not aware of any source which deals with this directly but it is standard manipulation of inductive and projective limits of locally convex spaces and their duals.