Hoare logic, proving conjunction rule from basic rules, possible or not?
Your reasoning is fine as far as the derivability of this rule. To respond to the latter part of your last paragraph, if the rules you've listed are all the rules available, then those are the only rules you can use. A derivation is a tree of applications of rules. You need to be able to state what rule you're using at each node in that tree. A(n explicit) proof that this rule was derivable would be an actual derivation, i.e. tree of rule applications.

There's a different and (slightly) harder question that you probably were not asked which is whether the Conjunction Rule is admissible. That is, could the Conjunction Rule be added without causing any problems, i.e. unsoundness.