Basically and briefly,
- Normalisation = transforming a data set to conform to a normal distribution. Converting to a frequency is NEVER a normalised transformation
- Standardisation = ensuring each point of the data set is comparable to the others, either directly or relatively depending on the calculation
The information you have received is incorrect, moreover you only convert to a frequency for a final assessment of the data and in my experience NEVER as an intermediate step. Normally data point between 0 and 1 are bad news and often require a transformation to get them away from fractions.
I don't understand the behaviour of the variance involved in metabolism and particularly lipids, however you should refer this issue to "cross-validation" Stackexchange. I did have some understanding of enzyme kinetics but in a different area.
Cross-Validation Stackexchance statisticians will understand less of the biology than I do (which is a bit worrying of course), so you will need to give them quite detailed information on the data set. They will almost certain support my concern of the advice you are given and provide a better method of standardisation. One method is to take the difference in mean and divide it by the standard deviation, you can then normalise the standardised data. They will want to know the goals of your investigation to understand who to process the data further.
What I can say is variation between runs is often an issue (in your apparatus perhaps not, I dunno) and standardising against a common control is one approach to overcome this, e.g. the control same common to all runs would be the mean about which all other data is centred.
What I suspect is that lipids are less popular than other fields, so standard protocol may not be observed.