No that's not possible. Yang means 'the one that/who is...'. In English you say to both 'yang' and 'bahwa' "that", so I understand the question, but in Arabic we also have 2 different words for it, so to me the difference is very clear. Yang gives an explanation or description of a noun (person, thing) mentioned before, where as bahwa explains the previous verb. The house THAT is big= Rumah YANG besar...you are describing the noun (house), so you use yang. I hope THAT the house is big= Saya berharap BAHWA rumah besar...you are describing the verb (hope), so you use bahwa.
I think you are asking whether you can use bahwa in this sentence. I want to know as well, it looks very applicable to me.