Whilst it may be the ST699 that is the problem, there could be other causes of your fault. Some boilers control the pump, so in that case you boiler may be at fault. Check to see if your boiler manual says that your boiler requires a pump overrun. If so, then the boiler will be controlling the pump.
With other systems that don't have a pump overrun, the pump may be controlled via an integral microswitch inside the the 3 port valve, that may be faulty and only operate intermittently. So whilst it maybe the ST699 that is the problem, it also may not be. Only electrical testing by someone that has a knowledge of the wiring of heating systems and the equipment to test it safely will be able to confirm for certain.
Regarding the swap, the wires required for the Hive are all present at the Honeywell, but you will need to look at the wiring diagram for the ST699 to see what each wiring terminal does and then find the corresponding terminal at the Hive that has the same function. [For example terminal 3 at the Honeywell is 'Central Heating On' at the Hive dual channel version terminal 4 is 'Heating On'.]
The ST699 will also have links between terminals L, 5 & 8. These are not needed with the Hive as it has these links made internally.
Finally the existing room thermostat must be decommissioned to prevent it interfering with the operation of the Hive. It cannot simply be disconnected otherwise the heating wiring will be 'open circuit' and the heating will never come on. This means that the existing thermostat switching wires will need to be bridged out by connecting them together at the wiring centre or other convenient point.