Not knowing which relo company you are dealing with nor the particular agent that you are working with, I cannot get real specific.
I will tell you that we have done several listings w/ relo companies and I have not seen an instance where they refused to counter on an offer that they thought was reasonable.
I have some question for you: is the agent you are working with a Buyer's Agent working for you, or are they working with you on the Seller's behalf? Is the listing agency for the property and your real estate agent's agency the same? Are you serious about buying this house or are you just looking to steal it?
Depending on your answers, I have a couple different pieces of advice.
1. If the Agent you are working with is not your Buyer's Agent and he works for the Listing Agency for the house in question, then you need to ask yourself if you trust him. If you do, then continue to work with him. If you tell him that you want to put in an offer, he cannot refuse and has to present it.
2. If the Agent you are working with is not your Buyer's Agent and he works for the Listing Agency for the house in question and you don't trust him, then you need to find a Buyer's Agent. He can help you find someone or you can ask around. If you go the second route, then your new agent is going to have to know everything you have done with this other agent up front... otherwise they will not get paid and you will have to pay it out of your pocket in the end.
3. If the Agent you are working with has no relation to the Listing Agency and you trust him, then tell him you want to put the offer in and sign him up as a Buyer's Agent.
4. If you do not trust the agent you are working with, then by all means, get yourself a Buyer's Agent. Again, make sure that they know the score at the start so that they can work out their commission so that you won't have to foot that bill.
5. If you are serious about buying this house, then put in a "fair" offer. Something that you feel reflects what you think that the property is worth, not the bargain that you are looking for. Sellers in general react badly when they feel that someone is trying to take advantage of them.
The thing with Relo companies and banks is that they are generally working on a formula of list price to sales price. They can not generally accept an extremely low offer unless the listing agent can justify it. If you are just lowballing the offer "just cuz" then you are probably wasting your time. Work with your agent to find that offer range.
The other thing with Relo companies is that the agents that list those properties will have to perform updated price opinions every couple of months. If they have kept the price steady since November, then that agent has given them a reason.
So... do a "gut check" on your realtor. Get representation if you feel that you should and then work with them to put together an offer that is fair. You do those steps and you should be moving in by the beginning of July.