How true is the original assertion?
The original assertion is poorly worded and not generally valid even if we ignore the wording. The assertion (at least a corrected version of it) can be true under certain condition for a limited period of time. However, the mechanism that the explanation talks about is not correct.
First, the statement is poorly worded, since precise language is important if we want be able to make logically sound argument we first have to address that. Capital in economics does not mean money, but actual productive assets, which are accumulated through investment. Hence, taken at face value the argument does not make sense since ceteris paribus more capital increases potential productive capacity of an economy which is deflationary (see discussion in Mankiw Macroeconomics 9th ed Ch 5). I do understand that whoever told you the argument likely meant money by capital but it is important to use clear language, especially since capital and investment will be mentioned further in this answer.
Second, what are we talking about is the wage-price spiral (higher wages leading to higher prices and higher prices to higher wages). Wage-price spirals do exist and can be empirically observed, but they are certainly not inevitable, in fact they are relatively rare and occur only occasionally. A quite comprehensive study of Alvarez et al 2022 could only identify 79 wage-price spiral episodes in sample of 31 countries spanning approximately 30-50 years (different countries have data available for different number of years). Some countries did experience no wage-price spiral and no country experienced more than 6 episodes. The only country that experienced 6 episodes was US which had the longest sample (of approximately 80 years) meaning that in US on average you would expect to see wage-price spiral maybe once every 14 years.
Moreover, all wage-price spirals in Alvarez et al did not lasted more than three months. Hence as you can clearly see these wage-price spirals are rather short lived phenomenon and they 'fizzle' out very quickly. So in real life they are extremely rare and certainly not inevitable.
However, real life wage-price spirals typically occur due to different reasons than employers voluntary committing themselves to offset inflation by higher wages.
Are my arguments against it solid, and do other/better arguments exist?
Some of them are some of them less so. As you correctly mention in your answer, workers could save and invest which expands the productive capacity of an economy and simply results in new macroeconomic equilibrium with both higher wages and lower prices. However, this is only partially correct argument because we know empirically that people do not have 100% propensity to save, and as a result higher income will always result in more consumer spending. Nonetheless, the argument above still could be applicable because deflationary effect of more investment and higher productive capacity can be simply offset by inflationary effect of higher aggregate demand. That is an empirical question.
However, your argument about shifting consumption pattern is very bad argument. CPI is based on a consumption basket of representative household. If people start consuming different items, prices of those items will rise (assuming there is no offsetting factor like increased productive capacity) and they will get higher weight in CPI calculation and thus contribute more to inflation. Hence, this is really non-starter. It could only work on paper if the CPI weights are not being updated, but it would just hide the inflation due to statistics not correctly capturing actual inflation.
As you also correctly point out if single employer does this it would have little to no effect on overall economy because no single employer is simply big enough. For example, in US the largest employer is Walmart, employing about 2.3 million employees. Yet this is less than 1.5% of US labor force. This is unlikely enough to such that Walmart alone could drive wage-price spiral despite Walmart already being outlier in having large workforce. This is probably best of the arguments you mention in your question. I think this is the best argument you can use regarding statement that talks about single employer, no complex economic arguments are necessary.
(Bonus question: If an employer felt it was "morally imperative" to not drive up inflation, then wouldn't they be duty bound to not increase their prices for the goods/services they provide?)
This is a moral not economic question so it does not belong on this site. However, a relevant thing is to note that many business are competitive. A perfectly competitive business already prices its goods equal to marginal costs so if wage costs increase prices have to increase in order to avoid economic loss. In less competitive industries there will be a markup, and business could technically lower the markup before incurring economic losses, but if mark up is 2% and wages increase by 3% then its impossible to lower markup to -1% without incurring economic loss. Business operating with economic loss is not sustainable and will likely shut down eventually.
Now because of Hume's guillotine we cannot get ought's from is's so you could still make a moral argument that employers could be 'duty bound' to not increase prices if they have some anti-inflation moral imperative, but fact is such duty would simply mean eventual shutdown of many businesses. A moral implication of such simple economic fact can be discussed on some site dedicated to moral philosophy.
