You can certainly ask for costs. However, such a request likely reduces the probability of your offer getting accepted. Judges rarely allow for costs on accepted offers to settle, because it is hard to figure out who "won".
Unless your costs are over $50,000 or so, I would just suck it up and not ask for costs in your settlement offer. In fact, even if your costs are over $50k, I still would not ask for costs in a settlement offer. Make the offer to settle, and then get costs if it goes to trial.