(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](http://en.wikipedia.org/wiki/A*_search_algorithm#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