Pediatric allergist

Yikes.  Please do NOT "sparingly" expose your kid to a known IgG or IgE allergen.  It does not build tolerance without other medications/treatments being involved (this is not the same as allergy shots which don't exist for food, or desensitization protocols which happen alongside drugs that suppress the reaction), and in fact can severely worsen the allergy. Every exposure in a shorter season gets a more severe reaction, and can actually aggravate the system enough if you don't get back to baseline that you fail other foods....which might actually be your problem here.  DO follow the trial protocol of single ingredient foods (none of the BLW they eat what you eat), 3 days on, then 3 days off with no other new foods, if no reactions, then it's a pass.  If you fail, you wait to get to 100% baseline (no rashes, reflux, etc.) before you try another food. 

If your child is truly having IgE symptoms (anaphylaxis, hives, swelling, vomiting, etc.), you absolutely need to fully avoid - you don't want to make an unnecessary ER trip right now.  If your child has IgG symptoms (upset stomach, bad diapers, rash - or FPIES - vomiting to mild or moderate shock), you need a GI, not an allergist.  Our son has MSPI and FPIES so our core team has been a pediatrician, GI, dietitian, and feeding therapist, and we've worked with an allergist to do a skin test to rule out IgE allergy. (Skin tests are both reliable and accurate under 2 - though kids can outgrow which changes the results and is not a sign of prior inaccuracy.)  If you are unsure of the difference, please consult the plethora of info on https://www.foodallergy.org/ re: IgE allergies, and www.fpies.org has lots of info on IgG intolerance allergies that cause rashes, colitis, vomiting, shock, etc.

If you need a great allergist, Allergy + Asthma medical group is great - I have IgE allergies personally and saw them when we had PPO insurance.  I felt like the rare adult there - their focus is definitely kids.  Just FYI that if your kid does show signs of increased IgG or IgE allergies, I'd highly recommend a switch to Kaiser Permanente.  It's life-changing to have your entire care team in one network, seamlessly communicating and sharing information, no need to transfer files or notes constantly.  And the cost control is incredible.  Allergy shots are a few dollars vs. a $20-30/co-pay at the PPO places.  Office visits are free to $20, and telemedicine is free.  Sameera Mian is a great GI, and Debbie Lin is a great allergist at Oakland Kaiser.

I highly recommend the pediatric allergy clinic at UCSF on 4th Street in SF.  We see both Laurence Cheng and Morna Dorsey and have had very positive experiences with both.  

My toddler (now 22 months) had a suspected egg allergy when she was ~8 months old and we were referred to Dr. Gendo at https://bayareaallergy.com/locations/berkeley. She ended up being diagnosed with a few other food allergies and they had us do some in office oral challenges and gave excellent instruction of how to safely introduce (or avoid!) foods in the home. 

I can also recommend Dr. Gendo! He did a really good job in explaining the pros and cons of avoidance versus hypo-sensitization therapy based on scientific evidence before he let us make the decision how we'd like to proceed. Like another user mentioned, we have also had a few in-office oral challenges so far (egg, dairy) and were always pleased with the care received.