Does a weighted Breadth First search have "memory" when moving to the next vertex?
How about transforming your original graph to a graph with dummy nodes and just use regular BFS on it?

The idea is: if you have an edge with weight $k$ between $A$ and $B$, remove it. Then add a path between $A$ and $B$ with $k-1$ nodes in between, where all the edges have unit weight. 

Now you can use regular BFS on your graph and it will reach the nodes in the same order as it would in the weighted version, without you having to transform the algorithm.

Ps. BFS always spreads out evenly, layer for layer like ripples in a pond. So if the next nodes to be added are A, B and C it will always add those nodes, but in which order does not matter- as long as the edges are unweighted.