I know the Gloster E.28/39 (hope I got that right) was drafted with a token (and optional) gun armament, but it was never really considered as a viable fighter (although in its final form, it appears to have been faster than a Meteor I). My understanding is that for all that it was only the second jet aircraft, it was a far more sophisticated airplane all round.
For that reason, I'm not sure an He-178 fighter would have been a going thing. By the time you turn it into one (not just adding or finding space for guns and ammo but also providing sights, radio, pilot oxygen, more fuel - endurance is lousy - restressing the airframe, altering intake and exhaust geometry and rearranging internal space to suit the required bigger, more powerful engine, etc.) you've quite possibly got an altogether different airplane in every way (one could potentially end up with the He162 three or four years earlier, which might well be worth doing). Among other things, the -178 just doesn't have the required performance; in late 1939, piston engined fighters in front line service are already nipping at its heels and that's just the gunless bare-bones prototype. Putting military equipment on it is only going to make things worse, and then there's the issue with the poor service life of wartime German jet engines.
It's a bit like the argument that the British generals mishandled the use of tanks in World War One. Perhaps they did, but the Mark One tank was never the Blitzkrieg weapon that the Germans used to overrun the Continent in 1939-40, nor was it ever intended to be - they requested something that could overcome barbed wire and machine gun posts, and that was exactly what they got and exactly the use they put it to. When they tried to make it do things modern tanks would find easy, it failed dismally.
Likewise, if you want your First Jet Aircraft to be a fighter, you've got to specify your desire for development potential from the start, and space & weight have to be saved for all the things a fighter needs, with all that this implies for engine power available. A conversion unit of He-178 A-0 fighters is IMO only going to disenchant the front-line pilots, which is the last thing you want to do in real life, although admittedly it might serve the purposes of your plot quite well!!
