Andy
First of all I have no proof as to which one is which, but checking a few sources this is what I can locate and draw conclusions from.
Both MR.1s were ordered under contract 87/A/865 dated 2.11.1916
Some official details obviously remain of A5177 since it was reported as "ready" in Feb 1917 while A5178 is not mentioned until Oct 1918.
Further more A5177 was allocated to CTD (Controller of the Technical Department) 25.10.1918, SOC 19.11.1918 due to corrosion, still known with CTD in Dec 1918.
Of A5178 I have no further details.
Not sure what CTD was, but probably a part of RAE at the time.
The first British historian who placed A5178 as the aircraft allocated A.58623 was J.M. "Jack" Bruce in 1957 or earlier (British Aeroplanes 1914-1918)
Even if the story in our Bristol Fighter book differ from British Aeroplanes 1914-1918 (Bruce place A5178 with the Aeronautical Inspection Directorate at Kingsway where it underwent static tests and got the number A.58623) both books say A5178 became A.58623
Checking Barnes Bristol Aircraft since 1910 both he and our Bristol Fighter book says A5178 was w/o 19.4.1919 by Capt Barnwell. That, to me, does not sound like an aircraft which has undergone static tests and looking at A.58623 doesn't look like an aircraft which has crashed and become a static test airframe.
So adding all things together, I find it much more credible that A5177, when SOC due to corrosion, went to Kingsway and received the odd "static test" serial A.58623. Only one photo, as far as I know, exists showing that serial number.
Just my two-cents thoughts
Cheers
Stig