I presume your BIM is also an E. coli strain that is resistant to your phage. It most likely will depend upon the mechanism of resistance and on the phage. For example if the BIM has deleted the receptor protein for the phage then it is highly unlikely you will be able to easily evolve a mutant that can use a different receptor (it is theoretically possible but will be very rare). On the other hand there are many other ways a strain can become immune to a phage that can be overcome by mutations in the phage. So it really depends.
In terms of infection mechanism, Phages can also be categorized into three types
(1) virulent phages always lyse the infected bacterial cell to release their progeny (2) temperate phages can either enter the lytic cycle as virulent phages or enter the lysogenic cycle in which the phage genome is retained as a replicable entity (prophage) for a long time within the host
(3) filamentous phages can attach to the sex pili of F+ Gram-negative bacteria and mediate production of progeny phage particles from the growing cell instead of lysing the cells, as the generated viruses are secreted through the cell wall (Duckworth, 1987). Both temperate and virulent phages can mediate transduction, although most successful transductions are mediated by temperate phages. The F+-specific filamentous phages can also mediate transduction of plasmid DNA (Ohsumi et al., 1978).