Ok, so situation with branding etc look like this:
There is company, let's call it X with own website X.com and they have shop oldshop.com. After years shop is well known in their niche, which is competitive like hell (you change something on your page and next day ALL competitors are doing the same).
Company X want to make their offer bigger, more mature so they want to open new shop, based on new technology, with new name (beacuse they don't like old one) on domain newshop.com and kill oldshop.com domain.
domain X.com and oldshop.com are quite well known, almost always in top 3 results are either oldshop.com or x.com, newshop.com thanks to many factors can't rank at this state. Last SEO destroyed engine, link, canonicals to the point that on test pages nothing index. I just finished making repairs but either way we can't compete with other brands on the market with this new site and 60-70% of revenue is from either shop or from calls that shop generates.
Killing it now would cut off this revenue. Since positioning this new shop is matter of months I want to save what I can and instead of newshop.com do something like x.com/newshop so x domain would carry on newshop and this would in theory minimalize revenue drop.
If not - separate domain newshop.com and then we'll wait for it to get revenue but with no budget at all it's near impossible to win with our competition.
Sadly I don't know owners very well but I have feeling that they don't have idea what they're planing to do and yet they don't want to listen. Personally, as marketer, I would leave oldshop.com alone and do redesign to make it more modern.