The service you provide involves three parties (principal, agent or customer) and hence you need to have robust agreements that make it clear to the client that you are an agent (disclosed agent) and in that case you only charge VAT on the £20 commission and the £80 is exempt. If not disclosed to the client, then you will be acting as principal, and VAT on the £120 is chargeable.
The other argument you can make is whether what you are providing is directly linked to the provision of dental services – depending on what you exactly provide!. However, per AXA Denplan VAT case, where AXA Denplan was merely arranging payments between dental customers and dentists argued their services were an exempt supply of payment or transfer services but rather the courts ruled that it was a standard-rated administrative service.