We (EC) have done this – the EC resolved to take legal advice when an owner threatened to take legal action against us. We were within the scope of the Regulations so that is/was fine, you need to get a costs agreement from the solicitor at the outset. Preliminary legal advice should cost well under the amounts set out in the Regs and then the solicitor can advise you from there as to what steps you need to take to ensure you comply with the legislation.
In our case we monitored the costs being incurred and put a motion to the next AGM in relation to the legal services that was passed, as the matter is ongoing.
It would be ridiculous if the EC couldn't do this as otherwise it would be in a position where it would not be able to respond to a legal letter or any threatened legal action. Note that there is a difference between getting legal advice and taking legal action. Taking legal action is pretty serious and can cost a lot of money, and an EC would be ill-advised to resolve to do that at EC level.
In this case it seems you want to overturn a resolution approved at a GM and that is different. I believe that you should go back to a GM, as the EC can't override that decision. Query whether it would be appropriate for the EC to seek legal advice in those circumstances. Another option is for specific owners to make an application to the CTTT, as distinct from the EC, the members of the EC can do that in their capacity as owners. You don't need GM approval for that.