It is normally the buyer's responsiblity to instruct their solicitor to carry out searches.
Barratt's part exchange with you will be "subject to contract" this giving them (and you)a get out if the search shows anything un-to-ward.
Barratt normaly set very tight time limits on their deals (as do most builders) requiring exchange within 28 days. By giving you additional responsibility, it makes it more likely you will fail.
Perhaps Barratt want to get out of the deal or perhaps cannot trust their own solicitors to carry this out within their time-frame.
All in all, it does seem a bit unusual.
Remember you can always walk away, don't let Barratt pressure you.