New Domain or Sub-domain

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Its better that you have the blog within your site. That way it will be more reliable and also easy to use for your users.

And if you have so much obsession with the domain then just buy it and then redirect it to the page where the blog lies.

Coming to your site. If you build a sub-domain then most certainly you have to do SEO work all from scratch as search engines considers sub-domains as separate domains. So, my suggestion to you will be that create a sub-directory under your site in this way you can benefit from the current link profile.