To take German language, let us look at how Heidegger talks about ‘being’: there is the ‘big Being’ (‘Sein’) and there is the ‘small being’ (‘Dasein’ meaning ’there-being’ or ‘so-being’). And then there is ‘Sein’ and ‘Seindes’. All these are translated by the word ‘being’.
To jump back a couple of hundred years, and pick up another philosopher with the same alphabet and the same language, we have G.W.F. Hegel. His ‘Sein’ of course has nothing to do with the ‘Sein’ of Heidegger, even though both are translated as ‘Being’ with a big B. So, even in one language, there are problems with respect to translating words into another language.
The post-world war English (and the formal logic of the twentieth century) assures us that ‘being’ is a synonym for ’existence’: if something has a being, it exists. Were we to use the word ‘being’ in this sense, then quite obviously ‘sat’ cannot mean ‘being’. For ‘maya’ exists and thus has ‘being’. But ‘maya’ is ‘asat’.
The ontological and philosophical assumptions of the English language into which one translates a word from another language and culture should make one wary of making bold claims. No study of any Sanskrit verb (from 11th century CE) can ever tell what that word means in English in the 21st century. The issue is not about Sanskrit verbs but what they mean in the English language of today. However, if one insists ‘sat’ in Sanskrit has the intension as ’to be’ in English, then, it follows that ‘Sat’ functions as an existential quantifier (’to be’, in modern English, is to be the value of a variable.), which it clearly does not. Before one can say ‘sat’ is “’to be in the sense of ’to exist’”, one needs to know the intensions of the words ’to be’ and ’to exist’. Since you make a distinction between ’to be’ and ’to exist’, may I know what these two words mean? What is their ontological and philosophical import?
I suggest that deeper investigation is required into language and philosophy before coming up with claims that ‘sat’ is ‘being’.