How to make a gel that doesn't harden at below freezing temperatures
There are a number of ways to depress the freezing point of a solution, the most common of which in an ice-cream context would be adding sugar and adding alcohol. Another solution, if you're not married to the idea of the lemon custard being a separate swirl mixin in differently flavoured ice-cream, would be to have the lemon custard be the custard for the ice-cream in and of itself.
For the first option, the higher the sugar and alcohol content of your lemon swirl mixin, the lower the freezing point will be, and the softer it will end up in the final product. The fat content of the lemon custard might increase and sort of 'spread out' the freezing point also. To this end, you could try making your lemon custard with a higher sugar content than usual and maybe some added vodka (or limoncello if you can get it) and seeing how that freezes. If it still freezes too hard for what you want, you might abandon the idea of a lemon custard and go with a lemon syrup or puree that might be closer to the consistency of more conventional berry swirls you see, while relying on the ice cream itself to provide the creaminess and fat content you expect from a lemon bar custard.
Serious Eats has a guide for making such a thing here.
You might also consider, instead of mixing in a lemon swirl, replacing the ice cream custard itself with something to evoke your lemon bar topping. I don't know what your recipe for lemon bars uses - I'd be familiar with a 'lemon curd' mixture made from lemon juice, zest, sugar, eggs and butter, but you might be using something more 'custardy' with cream and/or milk as the dairy component. In either case, if you modify that recipe to proportions more typical of an ice cream base (presumably by adding or increasing dairy content), you can skip the lemon swirl entirely and just have a shortbread mixin.