I ended up turning the offer down a few months ago.
You are late for negotiations, and don't have anything to walk away to.
more than 3x what I currently make
That could be due to higher cost of living etc in the new city/country, so fix upon some expected salary first. Is 2x good enough? 30%? anything over x? Is it available in your current location for your experience and work? etc.
I'm quite content with my current situation.
Any offhanded technique will end up burning bridges, so don't use them here, as it will disturb the balance you want in life (location, contentment).
If I were to attempt to negotiate a raise, would that be something appropriate to bring up?
Try not to put it across as a negotiation. What happens in the case when employer declines? What happens in case employer thinks you are looking to leave? This could hurt career progression.
Do not use the word "negotiation". Use the word "discussion" instead.
if there's a way I could use the offer to my advantage at my current job?
Yes. Discuss, not negotiate. You should here want them to bump your offer out of goodwill, not pressure tactics. Avoid hard stances of any kind.
Keep your expectations simple here. Talk to your manager, and let him know of facts:
Focus on the good things about the job:
2. Feel motivated
3. Like the work etc
5. Loyalty to the company
6. Feeling undervalued
Finally, check if there is something they could do for you. A good manager will not want to lose the goodwill of a good employee, but everyone has their constraints.