I'm confused by the statement "cost to refactor a software become onerous" the cost of refactoring is a fairly linear thing. It's the cost of adding new features into a mess that grows exponentially. That's what you need to focus on. As a start, can you pull together stats on the development team performance over time? The management might be able to give you this information. Try asking for it.
Often management can see this happening but doesn't understand why. At the same time the development team starts talking about things like tests and refactoring. It's important to understand that from a business perspective, tests and refactoring provide no value added to the product. It's not a feature that adds or retains revenues. They can't advertise: "now with unit testing!" So from their perspective, productivity is lagging and your answer is to do more unproductive things. It's not going to be welcome, but it's necessary.
The point you need to be clear on this that these are things that are needed to keep costs under control. Try analogy: If you never change the oil in a truck, eventually it will fail and the costs to repair it will far outweigh the cost of proper maintenance. Railroad operators don't like paying for track maintenance (it's pure cost!) but if they don't they don't have a business. The Pony Express had a fresh horse every 16 kilometers for their riders. It wasn't because they like owning and caring for lots of horses and maintaining so many stations, it was so they could be really fast.
You need to make them understand that what they are doing is like trying to run a pony express but by flogging a single horse the whole way. It's going to get slower and slower and eventually stop moving entirely. Once you get them to understand the idea, you need to explain that the horse is almost dead. If they don't invest some time in rehabilitating it, they will never be able to keep up with their competition. You are not asking for time to tidy up. You are asking for an investment in time so you can make things move fast again.
If you need a historical example to provide them with, you need look no further than Microsoft. I recall reading an interview with one of the top development managers about how all that mattered was adding new features and cleaning up old code didn't matter. It was boastful, they were so much smarter than their competitors by ignoring about the pile of manure the were building upon (I tried to find this but it was a long time ago and I can't remember the name. I want to say it was Allchin but I'm not positive. Any help?) IIRC, this was right around the time of Longhorn which was to become Vista. It's not too hard to see how those choices lead to MS losing it's dominance to it's competitors.