Method A allows fitting only one type of function to all pair of variables.

That sentence is wrong as written. The suggested method of correcting it is syntactically valid but, as you mention, it can change the meaning of the sentence.

Rather than writing each pair of variables, I suggest the following:

Method A allows fitting only one type of function to all pairs of variables.

All pairs treats them as a collective target.

Note, however, that there may be an additional issue with the meaning.

You say that you want to apply a single function to all of the variable pairs. But by using type, you're not making that explicit.

It's quite possible, for instance, that there are five different functions that are all of the same type. If so, it would not be the same function that would necessarily be applied, but only the same type of function.

If you want to make it very clear that you're applying one, and only one, function, then you need to make a further change:

Method A allows fitting only a single function to all pairs of variables.

If it's a single function, it is the same type by definition—since there is only one.

As for your second sentence, it's not wrong to turn it into a singular, but there should be a further revision:

Method B allows fitting a different function to each pair of variables.

Note that allows just means that it's possible. It doesn't mean that each variable pair can't have the same function fitted to it.

If you want to further specify that they are all of the same type, you can do that too:

Method B allows fitting a different function of the same type to each pair of variables.

Summary

The question relates to phrases of the structure kind of thing, and asks about accepted forms based on the plural forms of the words within the phrase. As observed, any uncountable noun may not appear in a plural form.

In every case, the original form of the phrase kind of thing is the accepted and expected form for the singular number.

Following are the three possible forms of the phrase that include plural forms of some of the words:

1. kind of things
2. kinds of thing
3. kinds of things

All three forms appear in modern English, as a plural form of kind of thing, and none is unacceptable.

Which, then, to choose, when needing a plural form in speech or writing?

Form (1), though found in respectable works, including those of Shakespeare, appears to have extremely limited use in contemporary speech and writing.

Form (2) is in active use, but appears less commonly than Form (3), which receives the greatest support from contemporary grammar texts and usage guides, and seems preferred by many native speakers.

A safe choice is (3), but (2) deserves better than to be dismissed.

Details

A discussion on the same topic, in the Stack Exchange community English Language & Usage, shows that the topic is complex and controversial, and any overly brief discussion may be inadequate.

(Compared to type, which appears in the original question, kind is a more common word in phrases of the general structure, and will be the preferred example in this discussion.)

Usage

Preference for kinds of thing over kinds of things may be more likely in cases of the following:

• An individual is British, rather than American.
• The context is a formal style of writing, compared to regular speech or plain writing.
• The word taking the place of thing is either of the following:
  • A noun that has common uncountable usages, even if it also has countable usages.
  • A noun that describes a topic in an abstract or technical subject.

Webster's Dictionary of English Usage gives an extended and complex discussion relating to the question (see entry for kind), which suggests that kinds of things may be the wisest and safest choice for contemporary writers, while also explaining that all of the other plural forms have been in widespread use as recently as the twentieth century. The Chicago Manual of Style (16th edition) apparently makes no direct mention of the question, but contains over one hundred instances of the general phrase in the form kinds of things, and none in any other plural form.

Contemporary usage, codified by respected style and usage guides, clearly prefers kinds of things.

Grammar

A Comprehensive Grammar of the English Language (Quirk et al), despite its rigorous treatment of grammar overall, makes only summary mention of the question (see section 5.6, Partitions in respect of quality), suggesting that kinds of things is a preferred pluralization of kind of thing.

Yet, the question appears to be poorly understood grammatically, and lack of agreement over any correct usage from grammatical rules suggests the occurrence of multiple grammatical interpretations.

In particular, ambiguity may arise, within a natural interpretation of the phrases, over the following two competing grammatical parsings:

1. [ kind of ] thing
2. kind [ of thing ]

In case (1), the preceding phrase (in brackets) functions as an adjective that modifies thing, the noun. Then, pluralization of the phrase requires that thing appear in plural form, with no clear requirement for kind. In case (2), the phrase following kind (in brackets) operates on it as a postmodifier, the same as the phrase of blue would do in sea of blue. Then, pluralization of the phrase applies necessarily to kind, but not so to thing.

Considering case (2), applying a plural form to thing may be inappropriate. In sea of blue, blue is an adjective, just as in blue sea. Similarly, because accepted grammar allows tree types, but clearly rejects trees types, the interpretation of the phrase of thing as a postmodifier suggests that tree, moved to the of phrase, follow the same pattern as blue, and retain necessarily the singular form.

Further ambiguity, however, arises because English usage allows either the singular or plural form for a word representing a category.

Consider the following openings of two Wikipedia articles, from the English versions:

• Elephants are mammals of the family Elephantidae and the largest existing land animals.

• The African elephant is a genus comprising two living elephant species...

If, in kind of thing, the word thing may be considered to represent a category, then a singular or plural form seems agreeable.

One advantage of kinds of thing is that it more closely follows the form required for uncountable nouns. Since kinds of rice is a plural form, and since kinds of rices is not on offer, kinds of potato might be preferred, as the world is filled both with much rice and many potatoes, but the measure of neither is relevant to the counting of their kinds.

Conclusion

The form kind of thing is the singular form of the phrase, whereas kind of things is considered plural, but is no longer common.

The form kinds of thing is an accepted plural form, and may be a suitable one, in many cases, but kinds of things enjoys greater support in contemporary usage and better favor from individuals currently. The latter is a safer choice, but some writers may prefer the former for a particular literary or formal style, or because of certain grammatical considerations.