One of the drawbacks of renting a house or an apartment, is having to live with pet restrictions.
Have the agent represent you in the search for new place. There is always way to convince the landlord to lease you the place. In addition to a contract, you should be offering to pay a damage deposit, or adding a substantial amount to an existing deposit. Your landlord may require this even if you do not offer. Offering one before he requires one just is a gesture of good faith.
If you need any help free free to contact me.