base verb or V-ing?

Solution 1:

• becoming a member of the House of Lords
• becoming a scientist
• keeping an eye on the government and representing the people

The foregoing phrases are all gerund noun phrases. They are not part of a to-infinitive:

Infinitives can be used as nouns just as gerunds are used as nouns. However, it is important to remember that the word "to" is included in the infinitive, whereas "to" is not part of the gerund, but is a preposition that comes before the gerund as a separate grammatical element.

preposition before gerund phrase or noun versus to-infinitive

Solution 2:

1.There are four stages on the road [to becoming a scientist], and I remember them all.

2.There are now a number of routes [to becoming a Member of the House of Lords].

3.They were supposed to be ordinary people who gave some of their time [to keeping an eye on the government and representing the people].

In these examples, "to" is not a subordinator but a preposition and thus the bracketed elements are preposition phrases with "to" as head and the subordinate "becoming" and "keeping/representing" clauses as its complement.