I was able to get GMock working correctly with the CppUnitTestFramework by creating a custom TestEventListener. I then created a simple set of interface functions to make it easier to work with.
I used the gmock 1.7.0 NuGet package to install the GMock framework into my project, and added the two files shown below. 
GMockUtils.h
#pragma once
#include <CppUnitTest.h>
#include <gmock/gmock.h>
namespace testing { 
    namespace GMockUtils {
        // Call once per test class or module to set up everything needed by GoogleMock.
        void InitGoogleMock();
        // Call once per test method to check for GoogleMock messages.
        void CheckGoogleMock();
    }
}
GMockUtils.cpp
#include "stdafx.h"
#include "GMockUtils.h"
using namespace Microsoft::VisualStudio::CppUnitTestFramework;
namespace testing {
    namespace GMockUtils {
        namespace {
            // Test event listener for use with CppUnitTestFramework
            class CppUnitTestReporter : public EmptyTestEventListener
            {
            public:
                CppUnitTestReporter() : _failed(false) 
                {
                }
                // Helper for converting std::string to std::wstring
                std::wstring to_wstring(const std::string& str) const
                {
                    std::wstring output;
                    std::copy(str.begin(), str.end(), std::back_inserter(output));
                    return output;
                }
                // Called after a failed assertion or a SUCCEED() invocation.
                void OnTestPartResult(const ::testing::TestPartResult& result) override
                {
                    // Log this result to the CppUnitTestFramework output
                    Logger::WriteMessage(result.summary());
                    // Note: You cannot do an Assert directly from a listener, so we
                    // just store the messages until CheckGoogleMock() is called.
                    if (result.failed())
                    {
                        _failed = true;
                        // Append this result to the running summary
                        _failedSummary += result.message();
                        _failedSummary += "\n";
                    }
                }
                // Clear any previous failures
                void ResetFailures()
                {
                    _failed = false;
                    _failedSummary.clear();
                }
                // Assert if any failures have been detected. Also resets failures.
                void CheckFailures()
                {
                    auto failed = _failed;
                    auto failedSummary = _failedSummary;
                    ResetFailures();
                    Assert::IsFalse(failed, to_wstring(failedSummary).c_str());
                }
            protected:
                bool _failed;
                std::string _failedSummary;
            } *_listener;
        }
        // Initialize the Google Mock framework for use with CppUnitTestFramework
        void InitGoogleMock()
        {
            // Avoids calling the function unnecessarily
            if (_listener != nullptr)
                return;
            // Dummy command line parameters (could pass exe path here)
            int argc = 0;
            char** argv = nullptr;
            // Initialize the framework
            ::testing::InitGoogleMock(&argc, argv);
            // We don't want exceptions thrown, regardless what the doc says
            GTEST_FLAG(throw_on_failure) = false;
            // Remove default listener
            auto &listeners = UnitTest::GetInstance()->listeners();
            delete listeners.Release(listeners.default_result_printer());
            // Create and install the new listener
            _listener = new CppUnitTestReporter();
            listeners.Append(_listener);
        }
        // Reset any previous failures detected by the listener
        void ResetGoogleMock()
        {
            _listener->ResetFailures();
        }
        // Prints messages and asserts if any Google Mock messages are found.
        void CheckGoogleMock()
        {
            Assert::IsNotNull(_listener, L"Google Mock framework not initialized by InitGoogleMock()");
            _listener->CheckFailures();
        }
    }
}
I use the GMockUtils functions in unit test classes like this:
#include "GMockUtils.h"
using namespace Microsoft::VisualStudio::CppUnitTestFramework;
using namespace ::testing;
namespace GMockUtilsDemo
{
    TEST_CLASS(GMockUtilTests)
    {
    public:
        TEST_CLASS_INITIALIZE(ClassInitializer)
        {
            // IMPORTANT: This must be called before any mock object constructors
            GMockUtils::InitGoogleMock();
        }
        TEST_METHOD_CLEANUP(MethodCleanup)
        {
            // Checks for GoogleMock messages. Asserts if found.
            GMockUtils::CheckGoogleMock();
        }
        TEST_METHOD(Method1_ParamIsOne_Method2CalledWithOne)
        {
            MockTestClass mockObj;
            EXPECT_CALL(mockObj, Method2(1)).Times(1); // Expectation will be met
            mockObj.Method1(1);
        }
        TEST_METHOD(Method1_ParamIsZero_IntentionallyFail)
        {
            MockTestClass mockObj;
            EXPECT_CALL(mockObj, Method2(1)).Times(1); // Expectation will not be met
            mockObj.Method1(0);
        }
    };
}
The Console Output
The console output now shows all GMock messages, and the run does not abort on the first test failure.
[3/27/2019 12:23:46 PM Informational] ------ Run test started ------
[3/27/2019 12:23:46 PM Informational] 
Unexpected mock function call - returning directly.
    Function call: Method2(0)
Google Mock tried the following 1 expectation, but it didn't match:
c:\...\gmockutilsdemo.cpp(64): EXPECT_CALL(_mockObj, Method2(1))...
  Expected arg #0: is equal to 1
           Actual: 0
         Expected: to be called once
           Actual: never called - unsatisfied and active
[3/27/2019 12:23:46 PM Informational] Actual function call count doesn't match EXPECT_CALL(_mockObj, Method2(1))...
         Expected: to be called once
           Actual: never called - unsatisfied and active
[3/27/2019 12:23:46 PM Informational] ========== Run test finished: 2 run (0:00:00.8631468) ==========
Test Explorer View
If I run the tests via Visual Studio Test Explorer, I can also see all GMock messages related to a particular test. It also works with the VsTest task on Azure DevOps.

Hopefully this will be useful to anyone who finds themselves in the same situation.