Higher risk-radiologist said not to have a mammogram?

Current models probably still suggest there is no absolutely 'safe' dose of radiation (there is no 'threshold' of safety-where below this dose there is no significant risk for cancer), but we are all exposed to some levels of radiation throughout our lives (if nothing else by cosmic rays).  The breast/breast buds of girls/women before age 20 may be at highest risk. There are different types of radiation, and that may be a factor also.

I know you tested negative for BRCA.  It is possible, though, as you probably know, that you could have some as-yet undiscovered single gene mutation.  Another complication is that at your younger age, most women have dense breasts, so mammography may not show anything anyway.

They do know that childhood radiation treatment to the chest (such as with Hogkin's disease) does put one at subsequent risk of breast cancer. There is an association with high doses of radiation (such as atomic bomb survivors, and other types of radiation exposure) and breast cancer. http://www.ncbi.nlm.nih.gov/pubmed/12105993

But you would be getting a much lower dose than that, of course, in a mammogram.  This study of people exposed to prolonged low doses of radiation (from Cobalt 60 in the steel reinforcement of their buildings)  showed a very low correlation with breast cancer (marginally significant correlation).  http://www.ncbi.nlm.nih.gov/pubmed/18666807

More to the point, this study, using many assumptions and models, opined that mammography is probably not of benefit for BRCA carriers in ones 20s. http://www.ncbi.nlm.nih.gov/pubmed/19176458   This European study found a risk for chest Xrays (which have much more radiation than mammograms) in BRCA carriers before age 40. http://www.ncbi.nlm.nih.gov/pubmed/16801631

In the UK, they do not normally use mammography for screening for women under 50.  This study opines there is some evidence that low-dose radiation may be more mutagenic than higher dose radiation. http://www.ncbi.nlm.nih.gov/pubmed/19454801  "Annual two-view screening should not beconsidered below age 35 years for women with no family history,and even for those who do have such a history, it should onlybe considered if the index patient was diagnosed below age 40years."http://bjr.birjournals.org/cgi/content/full/80/952/261

I am just giving you some examples of the data that the high-risk radiologist is using.

I think the evidence would certainly support the idea that MRIs would be safer (because there is so much less ionizing radiation exposure), but, as you say, your insurance will almost certainly not cover.

I'm sure there is controversy because of all the assumptions in these models. 

Its not an easy decision to make.