The risk you run is them losing interest in you by asking for a bump in salary. I'd guess 25% is a stretch, but it may not be. You could mention to them that you have received other offers and that theirs is on the low end. Ask if they are willing to increase the salary offer with a guarantee that you will accept it. You could ask for 22.5k, 20k, whatever. If they say no, tell them you would like a day to weigh all the options, then come back in the morning and accept. Tell them that after thinking about it, you believe that they are the best long-term fit for you.
I was in a similar situation a while back. I did not try to negotiate salaries (B's offer was only like 6% lower), but took B's offer. After starting there, it was clear to me that they wanted me bad enough I could have asked them to match and they gladly would have. That is not the case everywhere, however.
I'm just worried that they might think I'm lying about another offer to get a quick salary raise.
This is likely a common concern. Think about it from this prospective, however. Do you want to work for a company that assumes their employees are lying to them? Most likely they know what other companies in the area are offering for starting salaries. They also know who is hiring, how many open positions there are, and how many applicants there are. So they are likely to believe you.
I understand that I'm a graduate so it can be risky which is why I'm asking for a raise after the probationary period and not straight away.
I agree with Brandin's comment, the probationary period is for them to manage risk.
