Is doing this worth the time investment?
This cannot be answered by someone who doesn't know the context. There are many unknown factors here:
- How critical is a regression? Can faulty production code kill someone, or is it at worst having to send an email to have a bug fixed?
- How easy is it to change something in the codebase without breaking it? In other words, what are the odds of regressions popping up?
- Is development/maintenance ongoing, or are you near the final delivery?
Cost/benefit analyses are best left to management (albeit that I do expect them to ask for observations from the developers). You, as a developer, are not authorized to make decisions that affect the company's budget allocation or financial future.
It seems management has chose to not invest time in safety padding. Maybe that's a naive choice, maybe that's an informed choice. Who knows? Management does.
At best, you can pick it up with them and ask them if they have considered the risk of having to deal with regressions that turn development into a whack-a-mole game.
But in the end, the decision rests with them.
The alternative is to do try to develop carefully, functionally test, and pass it on to QA and hope for the best.
If management's decision is final; then you can mention that this is the only way you feel confident about further development in order to avoid time wastage on regressions. This again is a decision that management has to make: do they sign off on the slow-but-steady approach you're going to be taking, or are they explicitly telling you to rush it?
Very simply put, the responsibility of the decision lies with the decision maker. If management overrules your professional opinion, so be it. Do what they tell you to. And if it blows up, do not accept the blame for a decision that you argued against specifically so it wouldn't blow up.