Let's start with the past perfect:
"companies which had allegedly engaged in suspicious transactions"
Past perfect compares an action in the past which is completed prior to some other past point in time (which can be specified but also which can be implied). See:
past per·fect /ˈˌpast ˈpərfəkt/ adjective
1. (of a tense) denoting an action completed prior to some past point of time specified or implied, formed in English by had and the past
participle, as in he had gone by then. - Google
In this case there is a simple past item, the Prime Minister disclosed on Saturday but, prior to this simple past item, there was an earlier past point or points where the companies had allegedly engaged in suspicious transactions.
Second, let's look at the past perfect progressive/continuous:
"companies which had allegedly been engag[ing] in suspicious
transactions"
please note the form changes from engaged to engaging
The PAST PERFECT PROGRESSIVE TENSE indicates a continuous action that was completed at some point in the past; more precisely, something that started in the past and continued up to another action or time in the past (or else had finished just before another event in the past).
The change in meaning between the past perfect tense and the past perfect continuous tense is that, while the past perfect tense indicates that the companies had engaged in suspicious transactions at some points in the more distant past (but may have self-corrected that behavior so it could well be something that happened a few times 5-10 years ago but nothing more recent), the past perfect continuous tense would modify that meaning to indicate the companies had continued to actively engage in a pattern of suspicious transactions up to a point just before or even up to the point of the announcement by the Prime Minister.
Third, let's look at the past progressive/continuous:
"companies which were allegedly engag[ing] in suspicious transactions"
please note the form changes from engaged to engaging
The past continuous describes actions or events in a time before now, which began in the past and is still going on at the time of speaking. Both past continuous and past perfect continuous tenses can be used to talk about actions or situations that were in progress at a certain point of time in the past. While the past continuous merely shows continuity, the past perfect continuous tense also puts an emphasis on the idea of duration. It is mainly used to indicate the duration of a past activity or state.
In this particular case, as in many cases, changing to the past continuous tense would modify the meaning from the past perfect in much the same way that changing to the past perfect continuous tense would modify that meaning. The past continuous tense would modify that meaning to indicate the companies had continued to actively engage in a pattern of suspicious transactions up to the recent announcement by the Prime Minister.
Final notes on grammar
You will notice in a couple of instances above I changed engaged to engaging - it is necessary to use the participle form when using continuous tenses in active voice (passive voice is different). Think about a similar set of sentences:
People who had run in the marathon.
People who had been run[ning] in the marathon.
People who were run[ning] in the marathon.
Maybe using this similar but different sentence will help you understand why engaged needs to change to engaging in the second and third examples.
Edit
Because @ColleenV seems to be struggling mightily with this here is some additional information about understanding Active and Passive Voices in Past Continuous and Past Perfect Tenses. I hope the examples in this link along with the many I've already provided, including in my responses below, will help clarify this for anyone who is still struggling to understand it.
Past Continuous Tense
Active sentences in the past continuous tense have the following
structure:
Subject + was/were + -ing form of the verb + object
Passive sentences in the past continuous tense have the following structure:
Object of the active sentence + was/were + being + past participle form of the verb + by + subject of the active sentence