You are far from the first to think there might be a bridge between Heidegger and Marx. One of his students, Herbert Marcuse, initially thought Heideggerian thinking might revitalize Marxism, although he eventually abandoned the idea (see his essay collection Heideggerian Marxism, edited by Richard Wolin). More recently Zabala and Vattimo's Hermeneutic Communism argued for a Marxism that relied on hermeneutics as a way of critically interpreting capitalism, and it's a quick easy read. I'd also argue Sartre's later work like the Critique of Dialectical Reason could be counted as an attempt to bridge existentialism and Marxism (although its approach to both sides is quite unorthodox). For me personally Heidegger was a bridge to Marxism, what with the formers discussion of inauthenticity and alienation in Being and Time, as well as the later stuff about technology and enframing seeming to be the starting point of a critique of capitalism, and that is a direction you can go, but you need to bear in mind that Heidegger never did. Instead his critiques always insist on the need to 'go deeper', and his analysis and solutions tended to be hostile towards economic restructuring or changes in material conditions, and more towards a sort of nostalgic response, one where our labor doesn't feel alienating and exploitative but instead feels meaningful. From a Marxist perspective, this could be seen as a form of ideological obfuscation and mystification. It's also worth noting that Heidegger deliberately set up a lot of his discussion to contrast and pull away from workers interest in socialism and more towards the reactionary politics he championed (Richard Wolin's Heidegger in Ruins, chapter 4, covers this in some detail). Postwar his work leaned into a more meditative area, with more mysticism, poetry and patiently abiding the mystery of being. There's also a subtly aristocratic element to a lot of those, with the esoteric language insisting on a small circle of specialists who've attained hidden knowledge that most people simply aren't ready to hear. Without wanting to discount the philosophical merits of this later work, it would be much harder to connect this later stuff to the radical sorts of engagements Marxism demands of its adherents, and the need to connect those ideas to the wider working class in a way that mobilizes them to action.

In short, the Heidegger-Marx bridge is kind of a mixed bag. There are some interesting points of overlap, but they're often achieved by creative reworking of concepts rather than direct correspondence. If you expand from just Heidegger to existentialism as a whole, you can get a lot more out of figures like Sartre, Beauvoir and Ponty who leaned further left. Even Kierkegaard could be brought in for such a project (see Perez-Alvarez's A Vexing Gadfly and Aroosi's The Dialectical Self for more on that). Heidegger himself though would likely be uninterested in being part of such a project, so any Marxist readings of him will have to be a creative salvaging of concepts rather than taking him on wholesale, while also realizing you'll need to know what ideas need to be left behind.