Why are the fertility rates of large predators kept low?
This question can be tackled in several ways, and also seems to contain a couple of misconceptions about ecological processes (e.g. problematic group-selection ideas and how species 'fit' into the ecosystem). 
However, I think it is most usefully answered from the perspective of life history theory and the evolution of life histories (see e.g. Roff, 2002 and Fabian &amp; Flatt, 2012 for background), and at its core this question is dealing with tradeoffs between e.g. fecundity and mortality.

All organisms act under a limited energy budget, and when they spend energy on some activities that will limit what they can spend on other activities. 
This lies at the core of life history tradeoffs.
If organisms use resources to produce or raise offspring, this can therefore have consequences on growth, maintenance (e.g. disease resistance), predator avoidance and many other traits and behaviours, which all affect fitness.

With regard to your question, there is a well-known tradeoff between survival and fecundity (an intraindividual effect), which has been shown both between individuals within species and across species.
As with almost all species traits, there is also variability between individuals within populations, so that some have higher fecundity than others, but may then also suffer lower adult survival rates (maybe under certain environmental conditions). 
Therefore, there are certainly already "mutation(s) that increases fertility among the predators" within populations.
The reason that fertility is not continuously increasing over time, is that this would have negative fitness consequences overall.
So the reason that the (average) life history traits within a particular species are what they are, is that they are the result of the evolutionary pressures that the species have been experiencing over evolutionary timeframes, acting on the entire life history. 

To be sure, hypothetical mutations/changes in life history traits that would increase fecundity with no negative tradeoffs whatsoever for either parents or offspring (i.e. no negative fitness consequences) would definitely increase in frequency. 
From the perspective of life history theory, everything else equal, selection will maximize the survival function, $l(x)$ (survival to age x) and the reproductive function, $m(x)$ (number of offspring at age x), while minimizing maturation age. 
However, tradeoffs between these functions (directly or indirectly) and between generations will prevent these from being optimized one by one.

Population growth in long-lived predator species is usually also very sensitive to changes in adult survival rate (Caswell, 2001).
This result comes from analysing stage or age structured population models, and is the result of a life history strategy with high adult survival rates, fairly high age of maturation and relatively low fecundity. 
To increase fecundity in this situation with negative effects on adult survival will risk adult death and therefore loss of many future years of reproduction.
These kinds of results and models are often used to identify which life history traits and stages to target in conservation actions, but can also be used to understand evolutionary processes, since population growth rate can be used as an overall measure of fitness.

It should also be added that many other ecological processes must be considered.
For instance, large predators often care for their young for quite a long time, since their hunting/foraging behaviours are complex, and it takes a long time for young to learn how to hunt successfully by themselves. 
This means that adults need to bear not only the direct cost of producing offspring, but also to support them until they are self-sufficient.
For an extreme case, look at the Orangutan (not a predator, but still) where the young stay with their mother and nurse for up to 5-6 years (sometimes more). 
To increase fecundity in such a situation will put a large strain on adult mothers.
You also need to consider trade-offs between offspring number and offspring "quality", which have been demonstrated in many species (an intergenerational effect).
This means that an increase in fertility can result in lower offspring survival over longer timeframes, with overall negative effects on fitness. 

Finally, life history evolution is a large and complex field, and these are just a couple of examples and suggestions for why the situation isn't as simple as your question makes it out to be.
Environmental variability and the predictability of the environment are other factors that I haven't considered at all, but are both important processes when you try to understand life history evolution.
This question can also be approached from a more theoretical perspective, but without a background in life-history evolution I think it is more useful to start reading about different processes and tradeoffs, and then later if you are interested to go into the theory for a deeper understanding.