The challenge is that all budgets are locked down and the business owner is unwilling to take money from their marketing budget (or any other budget) to put towards analytics.
From my experience with clients and their budgets, it all comes down to what they can afford and what they are willing to afford. The reason why they don't want to invest in analytics is because they don't yet see the value in it.
I recommend trying to highlight how much faulty data can cost them. They could get the same results with half the marketing budged if they spent some time and money to properly track and analyse everything.
It's impossible to come up with good solutions by working with bad data. Also, if they have a 1% conversion rate it means they are literally wasting 99% of their marketing spend so how would they not want to know how to get more out of it?