If not for moral issues why is the government doing this? Why raise taxes on slave particularly? If the government needs more taxes and can get others to agree with it why not just a general tax hike on everything? In other words you need a reason for the Roman government to do this. Slave owners were a political force in and of themselves and you need a reason for the government to do this and to win against their opposition.
The reason is the problem indeed, if you insist on the condition not to overthrow the Empire. (For an attempt at a solution with an overthrown Empire, see TL in my sig.)
Religious reasons have been brought to the fore (Judaism), but they`re really implausible.
Economic reasons are beyond the grasp of Roman politicians.
Social reasons - e.g. much more frequent slave rebellions with much greater damage to important cities, in the best case to Rome itself - may be viable. But I suppose they might just have led to the manumission / emancipation of certain predefined groups at a specific point in time, not to an outlawing of slavery, which would mean that the situation could easily come back after a few decades.
Perhaps a combination of the first and the latter works. But then you end up with something that only faintly resembles the Roman Empire.
Also, and perhaps more importantly:
If you want to leave the general socio-economic outlook of the Roman Empire otherweise unchanged, then abolishing slavery won´t bring about a solution for the economic conundrums you listed. In the 3rd and 4th centuries CE, there were considerably fewer slaves than in the 1st century BCE. That didn`t solve the economic stagnation problem in the slightest. Instead of slaves, unfree coloni / indentured servants now worked on the latifundia after Diocletian´s reforms. Nothing much changed. And indentured servitude lasted for more than another millennium, in many countries well into the 19th century. That´s not because another economic outlook was impossible; it´s because power was concentrated in the hands of few and those few moulded laws, jurisdiction and the like as it best suited them, and even those who had no stakes in the business just couldn`t imagine things to be otherwise (and if they did, they`d get eliminated one way or another).
Overcoming mass slavery without replacing it with something very similar means overthrowing the socio-economic structures of Roman society. To put it in Marxist words: The means of production need to be controlled by class-conscious labourers. Class-consciousness here doesn`t mean red-flags-1st-of-May-kind of stuff, but for example the idea that paupers can give each other credit through common banks etc., and to have faith that this will work. Because that`s what the propertied elites had: actual control and a deep-rooted sense that people like them were capable of what it took to run an enterprise and a country.
A possible reason to increase the tax could be political.
After Cannae, the prohibition on the recruitment of slaves into the army was lifted, what if the slaveowners lobbied against this, and the final resolution was that the republic couldn't recruit slaves, but instituted a tax on the purchase of slaves.
I have another idea, that would have to work in tandem with the policy below.
The introduction of the suggested "Common Bank". Probably based off the experiment of a
Populares Senator. Near his main home (say a few villages), he provides the initial float for the bank. Encouraging people to use it, and it leading to small successes locally, but more income for the Senator, he establishes some branch offices/other banks further afield - more money for him - and then some traders emerge between the villages that weren't there before, creating more wealth, and spreading the news of the Common Bank that enabled this to happen - the early seeds of a Class-Concious working and merchant class.
On the assumption that the region flourishes enough to be newsworthy, it makes people aware of the value of it - and the Senator pushes the Senate to fund the institution of dozens if not hundreds of Common Banks. Probably more able to persuade members to do so as private investors because it increases their income.
Now assuming that this policy works, and is effective across Italia, then we could see the development of an emancipation movement, not for moral reasons, but because slaves cannot own property - and as such much more likely to be happy to see an increase in the tax of slaves. But this would most certainly be based on the actions of the merchant and working class, and those richer who didn't rely on slaves - so probably the more urban aspects of society.