You can not, rules-as-written, gain proficiency in the poisoner's kit from your background, custom or no.
Not quite correct. The rules also say, "If a character would gain the same proficiency from two different sources, he or she can choose a different proficiency of the same kind (skill or tool) instead."
This option is not limited to "tools found in another background".
So, if you are a rogue and take the Criminal background, both of which give you proficiency with thieves' tools, you can then take the Poisoner's Kit for your background instead.