One possibility would be to create a custom ContractResolver that, when serializing a specific object of type TTarget, adds a synthetic ExtensionDataGetter that returns, for the specified target, an IEnumerable<KeyValuePair<Object, Object>> of the properties specified in its corresponding DynamicPropertyManager<TTarget>.
First, define the contract resolver as follows:
public class DynamicPropertyContractResolver<TTarget> : DefaultContractResolver
{
    readonly DynamicPropertyManager<TTarget> manager;
    readonly TTarget target;
    public DynamicPropertyContractResolver(DynamicPropertyManager<TTarget> manager, TTarget target)
    {
        if (manager == null)
            throw new ArgumentNullException();
        this.manager = manager;
        this.target = target;
    }
    protected override JsonObjectContract CreateObjectContract(Type objectType)
    {
        var contract = base.CreateObjectContract(objectType);
        if (objectType == typeof(TTarget))
        {
            if (contract.ExtensionDataGetter != null || contract.ExtensionDataSetter != null)
                throw new JsonSerializationException(string.Format("Type {0} already has extension data.", typeof(TTarget)));
            contract.ExtensionDataGetter = (o) =>
                {
                    if (o == (object)target)
                    {
                        return manager.Properties.Select(p => new KeyValuePair<object, object>(p.Name, p.GetValue(o)));
                    }
                    return null;
                };
            contract.ExtensionDataSetter = (o, key, value) =>
                {
                    if (o == (object)target)
                    {
                        var property = manager.Properties.Where(p => string.Equals(p.Name, key, StringComparison.OrdinalIgnoreCase)).SingleOrDefault();
                        if (property != null)
                        {
                            if (value == null || value.GetType() == property.PropertyType)
                                property.SetValue(o, value);
                            else
                            {
                                var serializer = JsonSerializer.CreateDefault(new JsonSerializerSettings { ContractResolver = this });
                                property.SetValue(o, JToken.FromObject(value, serializer).ToObject(property.PropertyType, serializer));
                            }
                        }
                    }
                };
            contract.ExtensionDataValueType = typeof(object);
        }
        return contract;
    }
}
Then serialize your object as follows:
var obj = new object();
//Add prop to instance
int propVal = 0; 
var propManager = new DynamicPropertyManager<object>(obj);
propManager.Properties.Add(
    DynamicPropertyManager<object>.CreateProperty<object, int>(
    "Value", t => propVal, (t, y) => propVal = y, null));
propVal = 3;
var settings = new JsonSerializerSettings
{
    ContractResolver = new DynamicPropertyContractResolver<object>(propManager, obj),
};
//Serialize object here
var json = JsonConvert.SerializeObject(obj, Formatting.Indented, settings);
Console.WriteLine(json);
Which outputs, as required,
{"Value":3}
Obviously this could be extended to serializing a graph of objects with dynamic properties by passing a collection of dynamic property managers and targets to an enhanced DynamicPropertyContractResolver<TTarget>.  The basic idea, of creating a synthetic ExtensionDataGetter (and ExtensionDataSetter for deserialization) can work as long as the contract resolver has some mechanism for mapping from a target being (de)serialized to its DynamicPropertyManager.
Limitation: if the TTarget type already has an extension data member, this will not work.