Usage of "Proud words from a weak stomach"

The phrase "Small mouth, big words" in the example given is casting self-doubt on the speaker (with his "small mouth") to make the bold suggestion (the "big words").

English -- British English at least -- doesn't have a strict equivalent, although there are expressions which express a similar concept:

With all due respect, ...
If I might be so bold as to suggest ...
Far be it from me to say so ...

...all of which acknowledge the speaker's position.

As an aside, "small mouth, big words" sounds far more likely to be used dismissively of someone else whom you don't think should be making the suggestion. It's an insult. The reason there isn't anything directly equivalent is that we don't normally insult ourselves.

You could consider using "if I may say so" which is:

used for introducing a personal comment, when you know that the person who you are speaking to may find this offensive.

[Macmillan Online Dictionary]

You could also consider saying "if I may be/make so bold (as to)" which is:

a ​polite way of ​asking for or ​suggesting something when you do not ​want to ​offend someone: If I may be so bold, you still haven't ​mentioned why you're here.

[Cambridge Online Dictionary]

You could shorten both of them to "if I may" and it is broadly used when you feel awkward to cut in any conversation or start a new conversation in front of your boss or senior.

If I may say so/If I may be so bold, wouldn't it be a better idea to install wireless communication systems in police station too?

Most English formulations of this idea run a serious risk of sounding obsequious. If you want to make a suggestion to a more powerful person it's hard to find fault with a simple -

May I make a suggestion?