In Singleplayer, the Handgun HUD's bullet count shows up as empty (flashing the empty clip) whenever any clicking is done within the player's inventory.
In Multiplayer, the client doesn't seem to be keeping more than a momentary handle on the Handgun's bullet count. Bullets only appear in the HUD for a moment following reload, and the gun 'flickers' into reload as bullets are fired. Any player inventory clicks (or random updates?) will cause a Creative-mode player to require a reload, regardless of (prior) bullet count.

I get this issue 100% of the time playing on a server. I have had no single player issues with the UI. In Forge 10.13.3.1408. I know this is an old bug post, but the issue really degrades the experience of an otherwise awesome item.

My experience: Using the inventory does not alter the UI display at all. Simply equipping the gun adds the flashing reload HUD. Reloading can cause the rounds to show up in the HUD for a fraction of a second, but some times it is so fast that I don't think they flickered in at all. The HUD immediately goes back to the flashing reload indication. The gun is actually reloaded. Because of this conflict in the state of the gun, I think this is also letting me fire faster in multiplayer than single player, but I cannot measure that well enough to know for sure.

I have no additional information to help reproduce this issue. For me it is 100% on servers and fine on local play. If I can be any assistance on this issue, I am available and can supply the private modpack I am using.

Edit: A config option to disable the HUD would be nice if I wanted to force people to count bullets or just to keep using the mod while HUD bugs are present.

Have never experienced this; I am confused at your explanation of the issue. Are you saying the magazine is empty while you're in an inventory? Or that any change to the entire player inventory causes the magazine to glitch out?

The Singleplayer problems might have been caused by another mod (which I'm going to try to look into), as I went directly to testing with just Reliquary (on both server and client) in Multiplayer. It may or may not be worth mentioning that this was tested with Forge versions 1180 and 1230.
In Singleplayer it was just the HUD that was affected upon player inventory changes (involving any inventory slot being clicked); entirely visual, and would update to where it should be when the Handgun was reselected.
In Multiplayer, it seems that only the server is aware of the gun's bullet count, since the HUD always shows the blinking empty clip, and bullets are still fired in spite of the client apparently attempting to reload (resulting in the reload 'flicker'). The Creative-mode-in-Multiplayer issue of immediately (presumably) running out of ammo in the gun is triggered in the same way as the Singleplayer display issue (barring other mod involvement).