Note 1st suspicion that there is a different referance number for both and an initial in one under father, apart from that identical.
Note my second suspicion, if you look at Frances's death info and presume that the info is all correct is says Agnes 3 which would have made Agnes born 1861.

Two different civil registration reference numbers means two different births being registered.

1864 like all other years in our calendar, occupies twelve months. Agnes being aged 3, when her Mum died 22 August 1864, could be turning FOUR during 1864, but after 22 August 1864, or already turned three between 1 Jan 1864 and 22 August 1864, OR the informant may have been making a BEST GUESS of Agnes age, or the transcriber may have MIS-READ the handwriting and it may be "5" or "8". Mary's age may be 18 months, or perhaps there's a mis-read there .... perhaps 10 months, or 10 years, or 19 months, or 13 months or 15 months.
I am surprised you are asking for background research and prepared to pay for that research. I am sure that the researcher's techiques will be of a similar standard to many RChatters who give their time here freely. I feel sure that one of the first things any researcher does in these times, is to GOOGLE the name of the person they are researching. If that occurs, then they will find this thread easily, and see where you are already up to.
Cheers, JM