A bit more research suggests that something has gone wrong with his medal records (not wholly unusual given the numbers). You are right that the card showing his RB and Lab Corps numbers etc shows the theatre of war as France but that looks like an error.
The medal roll associated with his 1914-1915 Star, to which he was later confirmed as not eligible, gives his RB battalion as 19th Battalion. I am now aware that 19th Bn was made up in late 1915 of men serving in the Supernumerary Companies of a number of Territorial Force infantry regiments, among them the Cheshires. Their task in UK was home defence, the protection of vulnerable points.
That explains his transfer from the Cheshires* to the RB more clearly.
The date to theatre in December 1915 coincides exactly with the first deployment of 19 RB overseas, not to France but to Egypt where they were to be used in a similar guarding and protection role, freeing other battalions for the fighting tasks.
There is another medal card prepared after the war when he applied for a medal (presumably the 1914/15 relating to his service in Egypt). This has his Labour Corps number and confirms he was previously 19RB. It is also useful in that it shows he was in 812 Company Labour Corps, which was in Egypt. That card is on Ancestry with the wrong Labour Corps Company number! (search S Evans 312/Labour 19th Rifle Brigade (TF). That company was formed in mid 1917 in Egypt in the Palestine Lines of Communication element of the Middle East Force, the same organisation as 19th RB was in.
As to the engraving. Men who served overseas first in the Labour Corps had Labour Corps on their medals, one who served first overseas with another regiment had the first regiment engraved. It had to do with who kept their initial records.
His ineligibility for the 1914-1915 Star, even though it seems he was in Egypt before 31 Dec 1915, may well be hidden in the (very) small print that excludes certain parts of the Egyptian theatre. The notes on the cards do not explain it. He is not the only 19 RB man later shown as non eligible. I will dig a little further on this.
MaxD
*As an aside, luckily it doesn't confuse things because we don't have his records but he will have had another number when he first joined the Cheshires, they all had their own numbering system.