Well, I'm going to be the bad guy here, I guess. I get what you want to do and what you're asking, and I believe you are asking in good faith. So I'll answer in good faith, with absolutely no ridicule or rudeness intended.
Besides the pathogens, suddenly bringing in a new adult bird can really upset existing dynamics of your flock. And then there's the rooster damage to the girls - ripped up combs, loss of feathers on their backs, even deep scratches and wounds on their backs. Some roos will form an obsession with one or two girls almost exclusively and breed them over and over again, even interfering with their ability to get to food or water. And roosters protect their flocks - a rooster which was calm and docile in his familiar surroundings may see everything in his strange new place as a threat - and that includes you and you other members of your family.
I think before you even begin to plan this, you and your existing flock would be better served to get some information about breeding, mixing flocks, hatching, and raising the new chicks. I'm a little concerned that you didn't even know if the chicks could eventually lay eggs, and that's not dumb on your part, but it does show a lack of information and you need that before you rent this rooster. It's not like you can just a toss a strange rooster in the pen, then start hatching chicks. It's not that simple. I have put three roosters in my freezer. They were raised together with the girls as chicks. But when those hormones started kicking in they all became aggressive toward the girls and toward us. And if there's one thing many of us have learned, it's that when you start counting on all of those friends and relatives to take the excess roosters produced off your hands, most of them will have a million reasons why they suddenly can't do it. Are you prepared to cull if you have to?
I'm not trying to be mean here. But if you want to do this, then please have a firm idea in your mind of what you need to do, how to go about it so you don't end up with terrified and possibly injured chickens, and make sure you have good information on the hatching, raising, and division of the future chicks. These are living, breathing, amazing creatures and they all have individual personalities and quirks. Replace that picture in your mind of adorable, fluffy little chicks and put a good dose of reality in there. If you've learned as much as you can and you still want to do this then by all means, take what you've learned and go forward. And I wish you all the best.