Can I transduce my cells with a puromycin-resistant construct as well as one resistant to blasticidin?

We do not have a formal protocol for this but if you approach it in a step by step method, it should work. This might slow your cell growth so select first with puromycin after introduction of your puromycin-resistant construct and then with blasticidin after introduction of your blasticidin-resistant construct. You will have to optimize and minimize puromycin and blasticidin concentrations as both antibiotics can be toxic.