In Google Ngram Viewer, the search terms that he be,that he shall be,that he should be,that he were show the preponderance of that he should be in both AE up to c.1904 and consistently and to the present in BE.
In 1904, in AE, that he be took over until c. 2006 as the most frequently printed phrase of the four. Around 2006, that he be lost sufficient frequency as to place on a par with that he should be.
Without evidence I suspect that the 1904 American phenomenon was caused by a tipping point being reached that was caused by immigrants from countries with a language with an active subjunctive in all tenses of the verb.
NB Google Ngram Viewer is not too good at distinguishing AE from BE, so the results must be seen in that light. An interesting test is to enter “color,favor” in BE – there should be no results…
In both cases, it can be seen that the entire idea of the subjunctive and the implied subjunctive has undergone a decline.
It is worth noting that
1 English has only two tenses: the past and the present. There is no future tense. “I will/shall buy an apple” is clearly an expression of a current intention.
2 The artificial rule of “use the mandative shall with I and we, and the intentional will with other pronouns” stems from the erroneous belief that shall was appropriate because, whereas you could ensure that I or we did something as one had entire command of oneself (and thus grant certainty), you could not do this with you, she/he/it, they as they had minds of their own and their circumstances might change.
3 It is debatable whether “It is intended” can front a mandative – it is more likely that the “should” implies a strong “ought to” or slightly weakened “must”, both of which allow for exceptional circumstances: e.g. “Drivers should not sound their horn.” Does not apply if you are being attacked by a bear. And is thus marginally better than “Drivers must not sound their horn.”
That said if we take the examples
It is intended that this be extended = The current intention is that, at some time in the future, there is a extension to this and that that extension completed by someone.
It is intended that this shall be extended = The current intention is that, at some time in the future, there is a extension to this and that that extension is completed by someone who I have the power to command.
It is intended that this should be extended = The current intention is that, at some time in the future, there is a extension to this and that it would be proper for that extension to be completed by someone who someone/I has/have the power to command.
It was intended that this were extended = The old intention was that, at some time in the past, there would be a extension to this.
It can be seen that the subjunctive has no mandative force, it is more a wish with the strength of the wish being dependent upon the speaker’s command, whereas the pseudo-subjunctive formed with shall/should does have that force.
(This can be seen in the famous subjunctive example “God save the Queen!” In this there is no mandative force as one cannot “command” God to do anything, you can only hope that He sees fit to do something in line with your entreaty.)
is there any meaningful difference between shall and should in this case?
Yes, for the “shall” version see 2 above – it carries almost absolute certainty and for the “should” version see 3 above.