Hi Bumble B
You say some the other siblings Mary Ellen, Harry, Herbert and William were baptised Pre 1881 in Mirfield then you say parents James & Mary moved to Halifax post 1881. Changing Religion from Anglican to Nonconformist movement was very common in the late 18th and very early 19th with the like of John & Charles Westley and Methodist being George Whitefield 1844ish but by 1881 most people had settle in their beliefs! Therefore its possible but not probable. If this was the case with the topography of Heckmondwike and near places of Liversedge, Batley, Batley carr, Mirfield, High Town, Little town, Robert town, Norristhorpe, Gomersal, Cleckheaton, Rawfolds, Hartshead and Stainscliffe & Healey all within then distance that James took to Mirfield church means there are many possible main chapels plus loads of little houses made into places of worship. On top of this Heckmondwike was part of Birstall mother church as Anglican just over the hill from Heckmondwike.
What is the broad aspect too conside first! is your known facts, possible future of James post 1879 /81 also reasons needed your consideration first is when did James move to Halifax and did he live somewhere enroute between 1881 and Arthur's birth 1887 still as an Anglican worshiper
Workhouse BMD's and possible private baptisms not entered into the church later are some possible reasons as you will know

trying to find a baptism Prue July 1837 is a main challenge need!! post 1837 bapts hold very little info only their religion more than the birth certificate would hold. With the above obstacles you have a major challenge.
However! being that Annie is your direct Granny I would advise trying to contact your other relo's first to see if they knew of their religion! if they were nonconformist or not

as going to church them days was a big thing! None conformist even more as Anglican faith wavered in later years.
The task even if you live near repositories! the challenge is a mighty one!!! even for the most dedicated.
Kew in Richmond London may have the answer.
Dobfarm