UK citizen, employed by UK firm, delivering consultancy services for a Danish company in the USA

Can I perform this work with an ESTA or will I need a B-1?

There's no need to get a B-1.

First, the obligatory terminology note: ESTA is a system that authorizes visa waiver program (VWP) travelers to board. The terms of admission to the US are governed by the VWP itself, which predated ESTA by a couple of decades.

Second, for a business visitor, there's no difference in the kinds of activities allowed by the visa waiver program and by a B-1 visa. The relevant part of the statute, 8 USC 1187, Visa waiver program for certain visitors is paragraph (a)(1):

(a) ...
(1) Seeking entry as tourist for 90 days or less
The alien is applying for admission during the program as a nonimmigrant visitor (described in section 1101(a)(15)(B) of this title) for a period not exceeding 90 days.

Do not let the word "tourist" alarm you; section 1101(a)(15)(B) defines visitors who come to the US "temporarily for business or temporarily for pleasure." The letter B that designates this paragraph is the source of the B in the B-1 and B-2 visa designations:

(B) an alien (other than one coming for the purpose of study or of performing skilled or unskilled labor or as a representative of foreign press, radio, film, or other foreign information media coming to engage in such vocation) having a residence in a foreign country which he has no intention of abandoning and who is visiting the United States temporarily for business or temporarily for pleasure;

The converse of this is that if you are not allowed to do what you propose with the VWP, you're also not allowed to do it with a B-1 visa. You would need some other sort of visa. In this case, however, it seems likely that your proposed trip would be allowed under the requirements for business visitors.

With the information you've given in the question, however, it's hard to tell for sure. You (or rather your company) may want to ask a lawyer with experience in US immigration law as it pertains to business visitors.