DolphinQt: Improve the earlier panic alert deadlock fix

If the purpose of calling SetFullscreen using RunAsCPUThread is
to make sure that the GPU thread is paused, the fix in ef77872
is faulty when dual core is used and a panic alert comes from
the CPU thread. This change re-adds synchronization for that case.

Source/Core/DolphinQt/Host.cpp

@@ -36,6 +36,7 @@
 
 #include "UICommon/DiscordPresence.h"
 
+#include "VideoCommon/Fifo.cpp"
 #include "VideoCommon/RenderBase.h"
 #include "VideoCommon/VideoConfig.h"
 
@@ -89,15 +90,40 @@ void Host::SetMainWindowHandle(void* handle)
 
 static void RunWithGPUThreadInactive(std::function<void()> f)
 {
-  // If we are the GPU thread in dual core mode, we cannot safely call
-  // RunAsCPUThread, since RunAsCPUThread will need to pause the GPU thread
-  // and the GPU thread is waiting for us to finish. Fortunately, if we know
-  // that the GPU thread is waiting for us, we can just run f directly.
+  // Potentially any thread which shows panic alerts can be blocked on this returning.
+  // This means that, in order to avoid deadlocks, we need to be careful with how we
+  // synchronize with other threads. Note that the panic alert handler temporarily declares
+  // us as the CPU and/or GPU thread if the panic alert was requested by that thread.
+
+  // TODO: What about the unlikely case where the GPU thread calls the panic alert handler
+  // while the panic alert handler is processing a panic alert from the CPU thread?
 
   if (Core::IsGPUThread())
+  {
+    // If we are the GPU thread, we can't call Core::PauseAndLock without getting a deadlock,
+    // since it would try to pause the GPU thread while that thread is waiting for us.
+    // However, since we know that the GPU thread is inactive, we can just run f directly.
+
     f();
+  }
+  else if (Core::IsCPUThread())
+  {
+    // If we are the CPU thread in dual core mode, we can't call Core::PauseAndLock, for the
+    // same reason as above. Instead, we use Fifo::PauseAndLock to pause the GPU thread only.
+    // (Note that this case cannot be reached in single core mode, because in single core mode,
+    // the CPU and GPU threads are the same thread, and we already checked for the GPU thread.)
+
+    const bool was_running = Core::GetState() == Core::State::Running;
+    Fifo::PauseAndLock(true, was_running);
+    f();
+    Fifo::PauseAndLock(false, was_running);
+  }
   else
+  {
+    // If we reach here, we can call Core::PauseAndLock (which we do using RunAsCPUThread).
+
     Core::RunAsCPUThread(std::move(f));
+  }
 }
 
 bool Host::GetRenderFocus()