All perfect praise be to Allaah, The Lord of the worlds. I testify that there is none worthy of worship except Allaah and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
If the woman is adult and mature, then she is the one who decides the issue of her bridal gift, not her guardians or any others, because the bridal gift is her exclusive right. Allaah, The Exalted, says (what means): {And give the women (upon marriage) their (bridal) gifts graciously. But if they give up willingly to you anything of it, then take it in satisfaction and ease.} [Quran 4:4] If she wishes, she may give up her right to all of it or part of it.
Ibn Qudaamah wrote:
"It is allowable for the woman to give up her right to the bridal gift or part of it, exempting her husband from that obligation, or take it in her possession and then offer it to him as a gift, and if she is competent to dispose of her money (i.e. she is a sane adult), then her actions are valid. We do not know of any difference of opinion among scholars in this regard. Allaah, The Exalted, says (what means): {...unless they forego the right...} [Quran 2:237] The pronoun 'they' in the verse refers to the wives. Allaah, The Exalted, also says (what means): {But if they give up willingly to you anything of it, then take it in satisfaction and ease.} [Quran 4:4]..." [Al-Mughni]
Al-Qurtubi wrote, "The scholars agreed that if a woman who is legally competent gives up her bridal gift and willingly offers it to her husband as a gift, then it is effective and she is no longer entitled to it." [Tafseer Al-Qurtubi]
Hence, there is no harm in talking with your wife so as to reach a mutual understanding about giving up her right to part of her bridal gift; and she has the right to accept or refuse your request in this regard.
It should be noted that the Islamic Sharee'ah encouraged reducing the bridal gift as one of the means to have one's marriage blessed. It has been narrated by ‘Aa'ishah that the Prophet, sallallaahu ‘alayhi wa sallam, said, "The best woman is the one whose bridal gift is the easiest to pay." [Ahmad]
For more benefit, please refer to fatwa 106818.
As for what was mentioned in the question, that the amount of the bridal gift must be big enough to cause a bit of pain to the groom (for security purposes), then this is a false claim, and it is contrary to what we have explained above.
Allaah knows best.