Yes, they do have registered trademarks. Do you think we should rebrand even if we both have different names (with only thw 'owl' part in common), different branding (colour, design etc) and a completely different owl mascot / logo?
What's the company's name? Can't give you specific advice without knowing this. Also, take a look at the company's accounts (buy a digital copy via companies house for £10) - see if they are even in a position to take legal action.
However, just bare in mind, even if you are 'right' and you are not in violation of their trademark, if they have the money there is nothing stopping them from taking you to court after they have tried amicable means to solve the dispute to no avail.
Just to give you an example, I learnt the hard way that even when you are right, it is often better to settle than spend thousands trying to fight - especially if the other party has deep pockets.
I had a legal dispute before where I was legally in the right - 99% chance of success in court, but we decided to settle as going to court usually means everyone loses but the lawyers. It's also pretty stressful. Pick your battles.
I see you have experience of the 'picking your battles' thing and that is exactly what we need to do. This has been very useful and given me valuable perspective. I'll research as necessary and definitely post back here as to how we proceeded.