回复：关于自己付H-1B相关费用

yes. because only the $750/1500 fees are mandated to be paid by employer but this fee is waived for your employer.
Then again, when a legitimate American organization 不太情愿给我办H-1B, it's almost always not about the money. A few thousand dollars hiring expense is really not an issue to them. So maybe communicate and explain to them is more effective than you offer to pay the expense.