How to show/save value of custom customer address attribute on address edit Magento 2

Question

I want to add a custom attribute on customer_address. I need to save the value in DB, trigger the value in frontend forms (new/edit address) and in backend also.

I used the following InstallData script in order to add a new attribute:

$installer = $setup;
$installer->startSetup();

/* @var CustomerSetup $customerSetup */
$customerSetup = $this->customerSetupFactory->create(['setup' => $setup]);
$customerSetup->removeAttribute('customer_address', "chamber_of_commerce");

$customerEntity = $customerSetup->getEavConfig()->getEntityType('customer_address'); //customer
$attributeSetId = $customerEntity->getDefaultAttributeSetId();

$attributeSet = $this->attributeSetFactory->create();
$attributeGroupId = $attributeSet->getDefaultGroupId($attributeSetId);

$customerSetup->addAttribute('customer_address', 'chamber_of_commerce', [
    'type' => 'varchar',
    'label' => 'Chamber of Commerce',
    'input' => 'text',
    'required' => false,
    'visible' => true,
    'user_defined' => true,
    'system' => false,
    'sort_order' => 150
]);

$attribute = $customerSetup->getEavConfig()->getAttribute('customer_address', 'chamber_of_commerce')
    ->addData([
        'attribute_set_id' => $attributeSetId,
        'attribute_group_id' => $attributeGroupId,
        'used_in_forms'=>['adminhtml_customer_address','customer_address_edit','customer_register_address']
    ]);

$attribute->save();

$installer->endSetup(); 

I used

'used_in_forms'=>['adminhtml_customer_address','customer_address_edit','customer_register_address'] 

but it seems that only in the backend form is added automatically and also saves a new value.

I manually added an input in order to show the attribute in frontend on new/edit address in CustomerAccount, but I cannot manage to get the value from DB or save a new value.

I think I am missing something and I would really appreciate if someone can take a look and give a hint.

I am using Magento 2.2.4

Update:

1. I used an observer for Magento\Customer\Model\Address in order to save the value for edit/new address from MyAccount page.

2. In order to get the value saved also from the checkout page, I used:

2.1. extension_attributes for Customer AddressInterface

<extension_attributes for="Magento\Customer\Api\Data\AddressInterface">
    <attribute code="chamber_of_commerce" type="string" />
</extension_attributes>

2.2. etc/fieldset:

<fieldset id="sales_convert_quote_address">
    <field name="chamber_of_commerce">
        <aspect name="to_customer_address" />
        <aspect name="to_order_address" />
    </field>
</fieldset>

2.3. plugin for Customer Address for beforeUpdateData function

<type name="Magento\Customer\Model\Address"> 
    <plugin disabled="false" name="vendor_plugin_quote_model_address" sortOrder="10" 
        type="Vendor\Module\Plugin\Customer\Model\Address"/>
</type>