Most likely not. A lot of modern Irish words are just bastardized English, which I'm guessing "béar" is. I heard somewhere that Béarla comes from the word for "gibberish", but I'm not sure if that's true.
It's not bastardised English. Not any more so than English words are just bastardised French. Languages borrow from each other all the time (some way more than others). If you have a language that doesn't borrow at all from other languages, odds are you have a dead language.
Béar is a loanword from English bear. Béarla is a native Irish word that used to mean 'language', and is related to the word béal which means 'lip'.
I doubt béarla influenced the spelling of béar: it's more likely that béar was modelled after English spelling.
Weird. It told me I was typing in English instead of Irish until I got it wrong!
While this seems to be a Grant's Tomb question, and I got it right, it would have been helpful to hear both the singular and plural side by side.