In the scenario you describe the only thing you can do is pad your estimates. Don't do this explicitly but if there is some objection to cost make the assertion

In the scenario you describe the only thing you can do is pad your estimates. Don't do this explicitly but if there is some objection to cost make the assertion that the usecase though straightforward has to contend with certain legacy issues. There are two outcomes (1) they trust you, and grudgingly accept your estimates. You address what debt you can and ensure padding doesn't increase the next time thus trust is maintained (2) they force a timeline on you, in which case when it's late or broken in production you can at least say you needed more time. There is a third more egregious possibility, that they get an alternative estimate from elsewhere or one of your colleagues and in my experience this is the principle weakness to standing your ground "we'll take our business elsewhere" and I am truly sorry for your trouble if that is the case. Hopefully you have secure employment and the whole mess then just becomes the problem of the person that undercut you. Of course, in all these situations you have to make a call based on your personal/organisational circumstances but I just want to say that ultimately being upfront, diligent and continuously demonstrating trustworthiness may not always pay off in each particular instance, but over the longer term as reputation accrues they do.