With an explosive capacity that can devastate life and property for a

Nice explanation above. In response to 'that' and 'with' questions, using 'that' does correctly refer to nuclear warheads, but the modifier after property is used incorrectly (as pointed out above).

The 'with' modifier is acting as a noun modifier. Noun modifiers need to sit next to or very close to the noun they modifier. In this case it's clear from a meaning and placement standpoint that warheads are being modified, not proponents of peace.

KW

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With an explosive capacity that can devastate life and property for a radius of hundreds of miles, proponents of peace from several Western bloc countries met to discuss the continuing manufacture and deployment of nuclear warheads.

A. With an explosive capacity that can devastate life and property for a radius of hundreds of miles, proponents of peace from several Western bloc countries met to discuss the continuing manufacture and deployment of nuclear warheads. "With an explosive capacity"...incorrectly modifies "proponents of peace from..."

B. Proponents of peace from several Western bloc countries with an explosive capacity that can devastate life and property for a radius of hundreds of miles met to discuss the continuing manufacture and deployment of nuclear warheads. Again. "Countries" or "nuclear heads" have explosive capacity that can devastate life?

C. Meeting to discuss the continuing manufacture and deployment of nuclear warheads with an explosive capacity that can devastate life and property for a radius of hundreds of miles were several Western bloc countries. Awkward constuction

D. Proponents of peace from several Western bloc countries met to discuss the continuing manufacture and deployment of nuclear warheads that can devastate life and property with an explosive capacity for a radius of hundreds of miles.Property with explosive capacity? Haha

E. Proponents of peace from several Western bloc countries met to discuss the continuing manufacture and deployment of nuclear warheads with an explosive capacity that can devastate life and property for a radius of hundreds of miles.Correct

IMO E.
The question starts with a modifier+ comma+ main clause
To get the answer easy, just revert the sentence --> [main clause] + [modifier] since preposition 'with' nicely fit into the sentence without any changes.
all other options are having modifier issue.

sentence itself wrongly constructed since the initial modifier describes about a warhead but main clause starts with 'proponents'

Official Explanation

E. The introductory phrase in the original (preceding the comma) properly modifies warheads. Only choice E makes this necessary change while retaining the intended meaning of the original.