100 + 100 ≠ 200
First off -- you do not need a 200A service to feed two 100A panelboards, provided the total load as determined by NEC Article 220's calculations does not exceed the service ampacity. This is a consequence of the how 230.90(A) Exception 3 interacts with 230.40 Exception 2 and your specific setup (bold running text mine for emphasis, italics theirs):
230.90 Where Required.
Each ungrounded service conductor shall have overload protection.
(A) Ungrounded Conductor. Such protection shall be
provided by an overcurrent device in series with each ungrounded service conductor that has a rating or setting not
higher than the allowable ampacity of the conductor. A set
of fuses shall be considered all the fuses required to protect
all the ungrounded conductors of a circuit. Single-pole circuit breakers, grouped in accordance with 230.71(B), shall
be considered as one protective device.
Exception No. 1: For motor-starting currents, ratings that
comply with 430.52, 430.62, and 430.63 shall be permitted.
Exception No.2: Fuses and circuit breakers with a rating
or setting that complies with 240.4(B) or (C) and 240.6
shall be permitted.
Exception No.3: Two to six circuit breakers or sets of
fuses shall be permitted as the overcurrent device to provide the overload protection. The sum of the ratings of the
circuit breakers or fuses shall be permitted to exceed the
ampacity of the service conductors, provided the calculated
load does not exceed the ampacity of the service
conductors.
Exception No.4: Overload protection for fire pump supply
conductors shall comply with 695.4(B)(2)(a).
Exception No.5: Overload protection for 120/240-volt,
3-wire, single-phase dwelling services shall be permitted in
accordance with the requirements of 310.15(B)(7).
(Note that exception 5 has to do with the 310.15(B)(7) allowances for residential service conductor sizing -- they need to be taken into account when determining the total ampacity of your service and the ampacity of your individual sets of service entrance conductors, but stop there.)
A tale of a misplaced panelboard
But, there's more! While what the electrician did (putting a second main panel on your service) would have been OK when done properly as it would fall under 2014 NEC 230.40, exception 2 (bold running text mine for emphasis, italics theirs):
230.40 Number of Service-Entrance Conductor Sets.
Each service drop, set of overhead service conductors, set
of underground service conductors, or service lateral shall
supply only one set of service-entrance conductors.
Exception No.1: A building with more than one occupancy shall be permitted to have one set of service-
entrance conductors for each service, as defined in 230.2,
run to each occupancy or group of occupancies. If the
number of service disconnect locations for any given classification of service does not exceed six, the requirements of
230.2(E) shall apply at each location. If the number of
service disconnect locations exceeds six for any given supply classification, all service disconnect locations for all
supply characteristics, together with any branch circuit or
feeder supply sources, if applicable, shall be clearly described using suitable graphics or text, or both, on one or
more plaques located in an approved, readily accessible
location(s) on the building or structure served and as near
as practicable to the point(s) of attachment or entry(ies) for
each service drop or service lateral, and for each set of
overhead or underground service conductors.
Exception No.2: Where two to six service disconnecting
means in separate enclosures are grouped at one location
and supply separate loads from one service drop, set of
overhead service conductors, set of underground service
conductors, or service lateral, one set of service-entrance
conductors shall be permitted to supply each or several
such service equipment enclosures.
Exception No.3: A single-family dwelling unit and its
accessory structures shall be permitted to have one set of
service-entrance conductors run to each from a single service drop, set of overhead service conductors, set of underground service conductors, or service lateral.
Exception No.4: Two-family dwellings, multi-family dwellings, and multiple occupancy buildings shall be permitted
to have one set of service-entrance conductors installed to
supply the circuits covered in 210.25.
Exception No.5: One set of service-entrance conductors
connected to the supply side of the normal service disconnecting means shall be permitted to supply each or several
systems covered by 230.82(5) or 230.82(6).
However, your electrician screwed up when he put the second panel on the outside of your garage, as that ruins the grouping required by 230.40, exception 2 and 230.72(A):
230.72 Grouping of Disconnects.
(A) General. The two to six disconnects as permitted in
230.71 shall be grouped. Each disconnect shall be marked
to indicate the load served.
So, in any case, you'll need to have the electrician move the second panelboard inside to a spot next to where the first one lives, or replace the first panelboard with an exterior unit that can be mounted next to where the new one lives.
Grounding Electrode System
You're going to have to install a grounding electrode system. This is typically done by driving two ground rods spaced at least 6' apart, and connecting the rods together using a properly sized bonding jumper. Then you'll install a properly sized grounding electrode conductor, from one of the ground rods back to the main grounding bus in the main panel.
Bonding Water Piping
You may be able (required) to use the abandoned underground copper pipe as a grounding electrode, if it meets the criteria specified in the code. If the water is supplied to the building through metal pipe, you'll also have to bond the metal supply pipe to the grounding electrode system using a properly sized bonding jumper.
If the plumbing within the building will not be metallic, you don't have to bond it.
Bonding Gas Piping
As for bonding the gas pipe, according to the National Electrical Code, metal piping systems that are "likely to become energized" must be bonded (250.104(B)). So if you have a gas water heater that has no electrical connection, then you don't have to bond the gas piping. If you have a gas fired furnace, then it's possible that you'll have to bond the gas piping.
However, the bonding jumper only has to be sized to the rating of the circuit that is likely to energize the piping. NEC also allows you to bond the piping to the equipment grounding conductor, of the circuit that is likely to energize it. Which means the gas pipe feeding the furnace, can be bonded to the equipment grounding conductor feeding the furnace. And if the gas piping is isolated from the furnace electrical (not likely to become energized), you don't have to bond the piping at all.
In the case of the gas/electric range, again you can use the equipment grounding conductor of the circuit feeding the appliance, to bond the gas piping (if required).
Best Answer
You must buy lugs either way. Do it right.
You can't double-tap a lug. You can buy "staircase" lugs that are 1-lug width but provide 2 lugs, but you can't torque one without removing the other. And they may not fit on your breaker. In that case, you'd use 3-lug Polaris connectors.
Regardless, these don't come in small-large-large, so you'll have enough for dual 200A connections.
Since you're stuck buying the lugs anyway, it's cost-neutral whether you tap the bottom or top of the main breaker (e.g. Service side or breaker-protected side). Tapping the service side is a) Very Super Mega Bad from a safety and Code POV, because it's sending power to the subpanel totally unfused, and even worse (if that's even possible) b) since the sub is main-lug, it has no protection whatsoever for the buses - another show-stopper. Other than that, Mrs. Lincoln, c) it will have worse issues with bending radius due to the smaller area there.
So "on the breaker-protected side" is the non-exploding answer.
The breaker protects the wires
Given that the cost of wire is not critical when only going a couple of feet, use wire rated for 200A, so the breaker protects this wire. (Failing to do so is a codevio).
As long as you tap the breakered side, and use 200A wire, the subpanel doesn't need a main breaker if it has 200A+ busing. Your baby panel doesn't, so it would need a 100A main breaker, plus likely another pair of 2-lug Polaris to step the 200A wire down to something that will fit on a 100A lug. Therefore, for this and so many other reasons, I recommend this baby panel go back to the store, and spend a few dollars more on a 200A-bused panel - main lug is fine.
Not least, you won't be back here going "My panel is full, what do I do now?" Spaces are cheap; regrets are expensive.
Oh, snap: 310.15(B)7
There's one other wrinkle.
Right now, all your service's wiring goes onto the mobile home via those wires. That qualifies you for an 83% friendly "De-rate" based on NEC 310.15(B)7, since NEC is basically calling this an extension of your service drop, and letting you use service-drop sized wires.
Once you split it to two different destinations, you lose your 310.15(B)7 discount. That means adding this subpanel may make your long-existing cabling to the RV too small.
First, check if that is so. If it is, this is where you'd want to throw yourself on the mercy of the local permitting authority, and ask them to "grandfather" it.
And if they won't, then you could pull the meter, replace that 200A main breaker box with a 200A main-breaker service panel (as small as you can get it), add a 150A? 175A? breaker feeding the mobile home on its existing wires. Then, this "main" panel will have gobs of extra spaces for whatever you're trying to do here.
Heck, a wise choice of panel might even make a generator interlock both feasible and affordable.