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(The following may well have occurred to you already, but for completeness ...)

If I've fully understood the information that you have (and in context of A* pseudocode as you cited):

1. At any given time, the set openset holds the nodes that are candidates to step off to. Each of these nodes has an f_score[] value, which is the lower bound you refer to in your question.

(Intuitively, if I'm using A* to find the shortest path on a simple four-connected grid, my lower bound distance is the 'as the crow flies' distance, which is a lower bound to the 'follow the grid' distance.)

2. The condition you add is that, from another source, you know that the distance from $x$ to $dest$ should be no more than some upper bound.

Consequently when adding nodes to the open set (openset), you could ignore nodes that have a distance to $dest$ that are greater than your supplied upper bound. As in

upperboundDistToDest = (Calculation of upper bound)

(... then in the appropriate place ...)

if neighbor not in openset and dist(neighbour, dest) < upperboundDistToDest 
    add neighbor to openset