Cloning genes that make too much proteins

[Minsoo Yoon]

Dear experts:
I have a problem of cloning a fragment that includes a very strong
promoter. The fragment seems to produce too much products that become
toxic to cells.  It is getting more and more difficult to isolate
desired clones as the constructs become bigger and more complicated
with more components.  Unfortunately I'm using a PUC-based plasmid but
changing the plasmid to a lower copy one would be my last option. 
Could anyone in the know please advise me on this.   TIA