I think you can have a lovely day, especially if you plan it in advance.
I'd start with a clean and tidy house, with lots of fairy lights to counteract the gloom, short days and poor light of December. And some beautiful fresh flowers or a flowering plant as a focal point.
A long dog walk would be perfect, go somewhere social like a park or round town where you'll see people. Back for a delicious treat-like breakfast for you and your dog.
For me, I wouldn't do a massive cooked lunch, just a cheese and deli meat platter - to share with the dog if I'm honest. I'd get some really nice fruit as well, as a change.
In the afternoon, I'd walk the dog again to look at the lights (and exhausted families) then I'd come home to a hot bath. New PJs and then I'd sit down with a project - could be a new bullet journal, a jigsaw, a new diary, needlepoint or whatever takes your fancy. A new meditation app? Facebook keeps showing me a course on how to paint seascapes. I think the short days of December are great for planning for the turn of the year, looking ahead to longer days. you could pick some bulbs for pots or your garden?
A nice bone for your little dog and something delicious for you.
Sounds lovely - but you'll have your own ideas on what works for you. I don't think your day will be sad or lonely at all. Post a festive pic of your dog in a santa hat on social media to connect to others if you feel like it.