From the information posted, it seems to me that the situation is as follows:
1. Lauren has a legal liability to pay the charges.
2. Lauren's solicitor acted properly in obtaining an undertaking from the Vendor's solicitor to discharge the arrears of charge up to date of purchase. This can often arise where, for example, the bill for the charges is not yet available. If solicitors simply refused to complete purchases over issues such as this, commerce would grind to a halt.
3. Lauren has a comeback against the vendor's solicitor if he doesn't comply with his undertaking. He in turn has a comeback against his client (the seller), for whom he gave the undertaking.
4. Independently of the above, Lauren has a comeback against the seller under the terms of her purchase contract, but there is no need to enforce this contract, because a solicitor's undertaking is more readily enforceable.
5. Lauren's solicitor may well be entitled to charge extra for his time if he has to enforce the undertaking. The usual fee charged for a conveyance is predicated on the assumption that the contract will be honoured and undertakings complied with. Enforcement of a contract or undertaking is extra work. If the sum involved is small, Lauren might well decide to just take it on the chin.
6. Lauren can probably postpone paying the arrears if she explains to the management company that she is pursuing the previous owner for them, and that payment is secured by a solicitors undertaking.