That could occur ig, I was just assuming pein's dog vs pein's girl though
Thought it was neat since i've been doing summoning vs minion type matches lately eg danzo's summon vs danzo\s subordinates or jiraya's student vs his summoning
If Nagato does do that, then the fight would be a lot more difficult. Konan could try to outlast him by sustaining her paper form, but she wouldn't be able to just suffocate the dog anymore.
In these fights, it usually comes down to who breaks the rule. For Jiraiya, his summons are very strong because they exist independent of him and are more like teammates, while students are based on him and would usually be weaker than him. For Nagato, Konan is more like a teammate and not someone he groomed under him so her power is more likely to be comparable to his own than a summon that is based off of him. This is just a general trend.