I would too, if the market was slow and there were an abundant of properties on the market. But there's not so I wouldn't risk the additional wear and tear. It's a business after all.

To answer the original question, No. The Landlord can choose who they want in THEIR property in consultation with the Letting Agent. They do not have to give a reason, why should they? Its their house.
Your reasons make complete and utter sense...to YOU. The Landlord is still getting their same money from you as a couple as they would from you as a couple plus 1, so why risk any hassle and additional wear and tear. Regardless of your or your potential house share person professional status or age, you were agreed to let by the landlord based on YOUR credentials, now you wish to change all that so why should the Landlord bother.
All you are doing now is pushing it and creating unnecessary time on the Landlord and Letting Agency's behalf, when really just accept the house you are in and enjoy the life there, if its unacceptable to you then leave when your lease expires.

Although you do sound like a sensible, reliable tenant, unfortunately other less reliable tenants may have already ruined it for you. As a landlord myself, every single tenant has sworn they are the best tenants ever, will stay long term, won't damage it, will treat it as their home, will pay on time, will not make noise or have parties blah blah blah .. and they have all proven that it meant diddly squat. So your landlord may struggle to see that you are different to other tenants, so I think trying to sway him by telling him you are a good person etc is probably not going to gain anything for you.

As it stands, it sounds like the landlord has nothing to gain .. you have already signed the lease, and you are liable if your break the lease. Saying you will stay long term is not a tangible (or believable) benefit for him. So if you want to sway him, then perhaps you need to consider some tangible benefits for him, such as offering to pay a higher rent for having a third party added to the lease.

As you say the price for your house is now set. However in my experience, it is possible to gain a discount for having less people in the house than expected (I have done this before) .. so the time to negotiate was BEFORE you signed the lease, as your landlord was probably delighted to get a couple rather than 3 house shares. But unfortunately that is not any use to you now but maybe that will help you in the future when you are negotiating leases.

Anyway, I hope you do manage to find a resolution that keeps everyone happy.

Totally agree, and maybe offering more rent may be the only alternative which gives the Landlord incentive to consider it.

Thanks for the advice guys. I have a much clearer picture of both sides of the argument.

Im still waiting on the rental agent to contact the landlord again, they said it is a low priority. So I just have to wait.

They did say that if the landlord agreed to adding another person to the lease, this would amend the lease, and cost a weeks rent in their fee.

Smidsy, if you had asked us on here first, we would have said that your plan was likely to hit a few problems. The rental market works differently here, and it is very common for tenants to keep moving every 6-12 months at the end of each lease. Sometimes this is due to the tenants wanting a change, or their new house is ready, or whatever. Sometimes it is due to the landlord not wanting to keep them. Also, landlords will want opportunities to increase the rent, which can be written into a longer lease, or have more flexibility in renting/improving/selling the property, which could be very difficult with a long-term tenant. Having said that, your approach is worth a try, but don't be surprised if it doesn't work. It certainly wouldn't work for us.