Could the Feds objectives be adjusted to decrease or at least not increase wealth inequality?
Trivially, answer is yes. Congress could just give Fed a mandate for lowering inequality, the same way as congress could pass a law giving Fed mandate to bring world peace or solve world hunger or conduct space exploration. US is sovereign state and Fed is government institution and technically speaking US government can assign Fed whatever mandate it in its infinite wisdom chooses to.
Would changing the objectives of the Feds to: low unemployment, stable prices, and moderate long-term interest rates decrease or at least not increase wealth inequality?
No. Wealth inequality does not just depend on interest rates. We actually do not yet fully understand what is driving wealth inequality, since data on wealth (as opposed to incomes) is harder to come by and only very recently we became able to reasonably measure it (see Zucman 2019), but there is virtually no doubt that it depends on more than just interest rates.
For example, economic growth tends to reduce wealth inequality between countries but does increase economic inequality within countries (Zucman 2019). Next wealth inequality depends on design of tax system, capital incomes (Saez 2016) and other factors as well. So just keeping moderate interest rates would not reduce inequality or prevent its increase in itself.
Fed could probably do something about wealth inequality, if it would have mandate, but it would not be as simple as keeping interest rates 'moderate'. However, this would likely often conflict with its other two mandates and thus lead to worse economic performance, and hence the lower inequality might come at a cost of worse economic outcomes.
To illustrate this point, there is some evidence showing loose monetary policy during Great Recession increased relative wealth inequality in the US (Domanski, Scatigna, & Zabai, 2016). However, it is virtually unanimously agreed by macroeconomists that loose monetary policy is necessary during recessions. The reason why Great Depression became "Great Depression" is precisely because Fed decided to follow contractionary monetary policy (see Friedman & Schwartz Monetary History of the US). Great Recession of 2008/9, as bad as it was, was not even a half as bad as Great Depression if we look at metrics such as unemployment (during GD it was $\approx 25\%$ during GR $\approx 10\%$ see Petrosky-Nadeau & Zhang 2016) or fall of output (see here discussion of drop in US output during GD and here Fred stats for GR).
Lastly, there is generally a consensus among policy economists that technocratic economic institutions, run not by elected officials but experts and technocrats, such central banks, various regulatory bodies such as Securities and Exchange Commission (SEC) or Federal Trade Commission (FTC) should deal with technical economic matters and leave matters of redistribution of either wealth or income to elected officials who still have plenty of policy tools to, at least in principle, achieve any level of relative inequality their constituency desires. In principle arbitrary GINI coefficient could be achieved through arbitrary taxation and redistribution (although the same does not hold for absolute welfare and material standards). This being said there is no economic reason why technocrats should not be in charge of inequality this comes down more to moral philosophy and to belief that equity (as opposed to efficiency) related objectives should be decided democratically.
