Not entirely sure that I understand what your concern is.
Your formulae are OK.
the amount to add, offload or unload
These are just words for the same thing. You can be faced with two basic problems. In the first, you need to add or remove weight to achieve a CG. The usual buzzword for this is "ballasting". For ballast calculations, the significant factor is that the gross weight of the aircraft CHANGES when you ballast. Alternatively, if you are able, your preference should be to move stuff around to change the CG position. This has no particular buzzword term but the significant factor is that the gross weight DOESN'T change.
I understand the forward limit is non-linear thus these formulas will not work.
Not quite. The forward limit (as weight by CG) consists of two straight lines (ie linear), one in the low weight range, where the CG stays constant, and the other in the high weight range, where the CG moves aft. On the weight by moment (or IU) presentation (which gives exactly the same information, just reworked a little bit), these lines become, respectively a sloping straight line (linear) and a quadratic line (non-linear).
Looking at the formulae,
(a) for rearranging the load to change the CG, while retaining the same gross weight, you can use the formula as you know the CG.
(b) However, for ballasting, this gets a bit messy for the upper forward limit as you are changing the gross weight and the envelope limit is changing as well. There are various ways to go about running the sums but, for pilot folk, the usual approach is to plot the load change on the weight by moment (or IU) chart to get an approximate intersection from which you can figure out a reasonably accurate weight change required. For out-in-the-field work, flying one's aeroplane, one just makes this a little bit conservative and all is well. If you want to get the figure a bit more accurately (unfortunately, the examiner wants precision greater than is realistically appropriate), you use the graphical approximation to run a couple of arithmetic calculations to refine the answer to whatever precision you might desire.
In the lower forward limit section, providing that you don't get involved with the discontinuity where the lower and the higher sections intersect, you can use the formulae as the CG doesn't change and confuse the issue.
Could someplace clarify if there is a formula limit calculation, then what it is and how to use it.
Your question is a little confusing. What, exactly, are you looking for, here ? If you are after an equation to figure out the variation of CG with weight in the upper section, that is straightforward and can be found in the various textbooks. Unfortunately, the usually seen equation dates back to Bob Tait and Bob intentionally left it a bit more complicated than it needs to be so that the derivation sequence could be seen in the equation. That is to say, you can further simplify the derivation to make it a bit easier on the brain's memory cells.
I would like to clarify the exact situation in which an echo cog graph is required if there is no FL calculation.
The CASA workbook includes a weight by IU envelope for the Echo and a weight by CG definition in the text. There is no reason to look for a graphical presentation of the latter as you can get all the information you might need from the booklet as it stands. If your question relates to a graphical presentation of weight by CG, there is no requirement for you to have, or worry about, that format.
The graph seems quite time consuming like p charts
Providing you spend enough time practising with the various charts, there is nothing overly difficult with any of them. The more practice you do, the better and the faster you get playing with them when figuring out answers.
