I had a character who could use two light hammers, but for flavor reasons I wanted these to be light picks.  They would be pretty much identical to light hammers, but do piercing damage instead.

My first attempt involved just adding light hammers to my inventory and using the customization options to name them "Light Pick" and added the note "Piercing".  This was less than ideal because it still said it was a light hammer in the details, and rolling damage would produce bludgeoning damage in the game log.

My second attempt involved creating a homebrew "magic item" based on a light hammer, renaming it and swapping the damage type.  This made the damage rolls look right, but the details still said "light hammer" and mentioned the requirement of light hammer proficiency.  That doesn't matter for most characters who get simple weapon proficiency, but it might make a difference to characters with hammer or pick specific proficiencies or restrictions.  Probably because to make this work correctly there would need to be a light pick proficiency, and that would need to be in the appropriate collections.  The other annoying part is that you can't assign it a standard cost.

I then started down the path of customizing a war pick to make a "light war pick", adding light and thrown, and then reducing the damage.  Unfortunately, there didn't seem to be any way to change "martial" to "simple".  I suppose this gets back to the whole proficiency system, because war pick proficiency is probably a member of the martial weapons proficiency group on the back end, which doesn't seem to be something one can change.

So either way I am left to half measures, and my light pick is really just a sharpened light hammer.  I am using the second attempt version because at least it does piercing damage on the game log.

Does anyone else have any tips to solve these seemingly simple problems?

As off now you can't change the damage dice of the weapons nor change their proficiency so your stuck with that. You can choose to ignore some things or remove weapon properties. Eg. Rapier = Modifier = Ignore Weapon Property = Finesse. Also the part where you need light pick proficiency, if your using the war pick as a base weapon for your weapon. It will still be the same it will be still be proficiency in war pick. It wont change that unless stated other wise because the base weapon you are using is just a war pick. Usually to properly format a weapon its like this.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  

A bolter fires .75 caliber armor-piercing rocket rounds that penetrate a target then explode. It is a bulky two-handed firearm, with a pistol-grip and no stock, designed to be wielded by humanoids with enhanced strength. These statistics represent the Godwyn Mark Vb. Pattern bolter, the standard firearm of the Imperial Marines.

Burst Fire. While in range of its target the Bolter, Standard can instead fire within a 10 feet cube at a specific point, dealing 4d8 piercing damage on a failed saved, and half of that on a success, the targets must make a Dexterity saving throw DC 15 while in normal range, or DC 12 in long range. 

Only Creatures that are Large or Have a Carrying capacity of Large can wield this weapon with two hands.

It’s up to you to decide whether a character has proficiency with a firearm. Characters in most D&D worlds wouldn’t have such proficiency. During their downtime, characters can use the training rules in the Player’s Handbook to acquire proficiency, assuming that they have enough ammunition to keep the weapons working while mastering their use.                                                                                                                                                                                                                                                                                                                                                                                   So as of now you can't change the proficiency as well because of the base weapon. The base weapon I am using for this is a rifle, automatic which still uses firearms proficiency. So if you want a simple weapon light war pick choose a weapon that is a simple weapon. And adjust the damage Eg. dagger add another dice dice of 1d4 damage to 2d4 or 1d4 + 1d2 damage cause why not if you want a d6 damage type. I don't remember if you can remove the base weapon damage of the item your customizing. But If you find it remove the base damage and then add a new damage dice. So I think what you did is correct.

Your second attempt with the sharpened hammer is the best you can do. We are actually prohibited from creating new base weapons currently. It is impossible for you to make a “light pick” with its own proficiency. My “saber” is still just a Rapier that does slashing damage, my “arming sword” is still just a Longsword without versatile, there’s nothing to be done about it.

How did you add burst fire to this?

In case it helps, the dagger mechanically works the same as the light hammer (except for its finesse property, which doesn't change anything if you plan to use STR for attacks). All classes are proficient with the dagger, so proficiencies shouldn't get in your way. And it deals piercing damage, meaning it can be re-flavored as a miniature war pick.