Provided you have licensing insurance cover, I would fight it. However, Philpott has a point on co-terminous certs. That said it cuts both ways, if they regarded any of the incidents as relevent to your suitability to hold firearms, why would they regard it as irrelevent for your SGC and grant you a Shotgun Cert? Also, have theere been any issues with holding this? If not, then it seems to negate a lot of the reasons for denying you an FAC on safety grounds. An appeal does put the SGC at risk although question could they now revoke your SGC even if you accept a refusal on the FAC as they've discovered medical grounds that make you unsuitable? You need legal advice....
In my opinion on the points you rasied as a non firearms expert ie just a forum member:
Accidental overdose - should be irrelevent provided it was recorded as such. Anyone can accidentally mix tablets up. You need a copy of your medical records to see what your doctor recorded. You have a right to obtain these from your GP for free. It may be your records either record this abiguosly or imply or state suicide attempt. That could be an issue although it's long in the past. You need the records to ascertain the exact nature of what was recorded and whether it was recorded correctly. As it's impossible to know if something is accidental in many intances, it may have been ambiguos, which may be the source of your problem. eg. your records may simply say "overdose". Maybe if that's the case there is other evidence to show it was accidental eg A&E hospital records, family witnesses etc
Personally, I'd say to anyone having any kind of issue that may affect an existing certificate to watch what your doctor writes in your records and correct him there and then if incorrect or if he uses language that accidentally implies another meaning. Watch out for poor / inaccurate wording. It's far easier to get it corrected on the day there and then when it's fresh in his mind. than later when he'll probably refuse. If he writes it up after your visit, then request a copy of your record same day and challenge it immediately if wrong. If your doctor needs to know why your asking him to use different language, explain your are a licensed firearms holder and it may cause you problems if he puts that because it wrongly implies... You can even prempt him and remind him in advance of him writing the record that you are a licensed firearms holder and therefore can he please be careful of his wording because... If you are genuine and the matter is simply a matter of poor choice of words, most doctors will amend their language to ensure they don't give the wrong impression. Just don't expect your doctor to lie or cover up for you if you do eg actually do have a relevent condition!
Btw I believe the Firearms dept has no right to see your records, only to request a factual report from your doctor which isn't allowed to contain opinion. However, unsure 100% if this still holds true as I believe changes was debated. Again you need proper advice as to what they can request. Might be worth requesting a copy of the mdeical report as well, although many doctors will copy you in on their report. Depending on what the actual records say, your legal team may or may not want to use them as evidence. My advice would be don't say anything directly to the FEO's / licensing dept. Pass all enquires to your advisors.
Low Mood 2010 - My understanding is low mood is not the same as depression. Everybody suffers low mood at some time and low mood is not a depressive illness. I wouldn't have thought it was a reconised risk factor in itself although there may be some concern of it leading to depression if there have been more recent or frequent episodes.
2018 incident - this one is more tricky. However, the fact the CPS offered no evidence is in your favour. Your continued relationship might be an issue though because victim or aggressor, they may be worried of recurrent bullying and the possible loss of temper resulting in threats of or the use of firearms in response. It might be helpful for your legal team to try to find out why the CPS offered no evidence. I'm not sure if they can get them to disclose this. However, the best outcome would be if the CPS maybe they thought your use of force was reasonable.
My advice would be ensure you have insurance cover for licensing matters and if you do and they are willing to fund it and take it on, then appeal provided the co-terminus advice is favourable. It may mean you attending Court. If you do appeal, you need to direct everything through your appointed solicitors and not speak to the police directly, beyond saying you'll pass it onto your legal advisors. Some forces have anti's in Licensing departments who will use any excuse to deny you a licence including tricking you or provoking you to say something incriminating. They shouldn't be in licensing departments as there is a clear conflict of interest but some are. Never assume the FEO is neutral or your friend. They're not. They're there to do a job and some will use unfair means to achieve it. Be clear and honest in all dealings and think before answering. Don't get drawn into laughing and joking with your new "friend". Keep it distanced and professional.
On the calibre fronts, the FEO may have a point. Expanding ammo is unlawful on ranges except where sighting in a hunting gun. Isn't .17hmr manufacturerd in expanding ammo only? I can't comment on .300 blk out as although I know a bit about the round, I don't know the ammunition available. It maybe a good compromise might be a single gun at 1st of lesser calibre such as .22 rf with a view to applying for a larger cf calibre at a later date if cf target is your aim. A .22rf is a very common target gun and good way of showing trust. That said, atm, with a medical refusal, calibre is somewhat irrelevent. You need to overturn the refusal before even considering calibre.
BTW if you don't have licensing cover, be aware that costs can go sky high in these appeals. A single stage appeal could result in £10K costs, so you need insurance unless you are in a financial position where you have nothing the defence can claim costs against.
Can they still change the decision though? I understood once a formal refusal was issued (he should have the refusal in wriitng, usually hand delivered), it can only be reversed on appeal.