It looks like false logic to me too, but I'm very aware that that is how many Christians "prove" their religion to be true. 'The Bible says this historical/Godly event happened and this archeological evidence supports the account in the Bible, therefore the Bible must be true about everything so God exists and I'm going to Heaven.' Which sounds very similar to 'This is a part of what you say about your religion and it may be proved false one day, so your religion might be too.'

Is it okay to slip into the streams of thought that the other considers logic in order to beat them at it and potentially shake their beliefs?

I'm sure it's possible to believe in God but deceive oneself into belief of atheism. And then grieve shallowly with a feeling that the deceased is not really gone forever.